While JimeDorje gives a great answer, I think I can expand a little on how they interacted once they all entered China, since they grew in response to one another once they began to have conflict. I want to stress that JimeDorje is correct in what he says, so look at this as more of an expansion based on having read his reply first.

When Buddhism entered China in the 1st-2rd century CE, it was seen as sort of an alternative version of Daoism and the earliest stirrings in China used Daoist thought and ideas to convey similar Buddhist ones. The first mention we have of a community in China is in 65CE where one Prince Ying, brother to the Yellow Emperor, is mentioned as "esteeming highly the profound sayings of the Yellow Emperor and Lao-Tzi, and the virtuous deeds of the Buddha," indicating that Ying had, at least in some sense, converted to Buddhism or adapted the Buddha into the framework of Daoism. In any case, these early communities seem to have accepted or even encouraged these comparisons to Daoism and freely fused them. Legend even had it that the Buddha was Lao-Tzu who had gone to teach the "barbarians" Daoism or that the Buddha was a student of Lao-tzu who had gone to India, but misinterpreted the teachings. The idea that such a philosophy with a rapidly growing congregation was completely foreign scared the Chinese and Daoist elite, perhaps it was simply inconceivable to them that such a thing was possible.

In any case, by the third century Buddhism had become well established in China as a sort of "pseudo-Daoism", a syncretic mixture of Daoism and Buddhism. At this period, many more complicated Buddhist doctrines were muddied, simplified, or dropped all-together. For instance, the Abhidharmic and early Mahāyāna thought both, in different ways, denied the existence of a soul or self (ātman) instead preaching the idea of an impermanent combination of aggregates forming the idea of a self which was truly the not-self (anātman); The Madhyamaka school of Nāgājuna denied that any experience truly existed as we perceived it and everything was "empty" (Śūnyatā). In contrast, the early Chinese version of Buddhism taught an immaterial yet indestructible soul with a real effect on the mind. That said, it preserved the doctrines of Karma, rebirth, and escape through the quelling of passions; all of which were foreign (at that time) to the Chinese understanding of the world (Don't worry, this won't be on the test, but do remember it as it does become pretty important). The external similarities of the religions allowed for an easy mix and they were practiced alongside one another. Occasionally, they came into conflict, but it was rare.

By the late fourth century however, their was a significant push to remove the Daoist elements from Buddhism and establish itself in accordance to the Mahāyāna doctrine, in particular the Prajñāpāramitā sūtras of Nāgārjuna's Madhyamaka school. It was at this time that Fa-Hsien (better known as Faxian) went to India to retrieve the Vinaya (think of it as a set of rules in verse) and major sūtras. This increased the prestige of Buddhism in China but brought them into significant conflict with the Daoists and lead to a sort of revival of Daoist thought.

Before Buddhism, Daoism had no established canon or hierarchical structure. Works were written on it, some very influential and volumous, but compared to the wealth of sūtras entering through the Silk Road and the carefully organized, systematized teachings of Buddhism, they hardly seemed impressive. It was in the fifth century that Daoism first organized and systematized a canon and began a strong push for organization and writing sūtras for Daoism in imitation of Buddhism. Similarly, Daoism seemed not to have had a strong monastic or priestly component outside of shamans or village practitioners, but its rivalry with Buddhism changed that. The previously conservative tradition started building large temples and statues for the immortals and formal places to worship and practice in response, the quest for immortality was no longer something practiced alone or in small groups, but in a temple among others.

It was also around this time that Buddhism began to enter even the upper echelons of society in the Former Qin (not to be confused with the Manchu Qing of the 17th century to 19th century or the Qin dynasty of the 8th-3rd century BCE) dynasty and encounter Confucian thought. This proved a problem as the notions of subservience to the emperor weren't present in Buddhism like they were in Confucianism, so by patronizing it the Emperor ran the risk of having a competing power that was not subservient to him. However, the the fusion of ideals was made and the arguments that the Saṅgha was withdrawn from the world and thus did not pose a risk to the power of the emperor seemed to satisfy them in those days.

Under the Sui dynasty, Buddhism replaced Daoism and Confucianism as the state ideology and forced them once again to restructure in response. Confucianism had perhaps the most dramatic shift during it's resurgence during the anti-Buddhist T'ang dynasty and the subsequent Sung.

Neo-Confucianism was formulated in direct response and opposition to Buddhism and thus challenged its ideas directly. Of special interest was the idea of Śūnayatā which posed that all experiences were "empty" and there was thus no origin of things. Neo-Confucian thought directly opposed this saying that everything originated with Ch'i and all things would again become ch'i. There was also a marked criticism of the Buddhist ideas of meditation and emphasis on monasticism, with Neo-Confucianism considering it unhelpful to the development of society. By contrast though, the ideas of quelling the mind and contemplation became an integral part of Neo-Confucian thought and practice, something borrowed from Buddhism.

To jump back a little in time, in the 6th century, the two most influential schools of Chinese Buddhism developed, Pure Land and Ch'an (better known as Zen).

Ch'an incorporated the aniconic simplicity of the earlier Daoist ascetics and emphasized a nontextual tradition focused on personal self-improvement via spontaneous enlightenment. Ch'an also managed to survive persecution under the T'ang and Sung due to the fact that it held a rigorous standard for monks to work and contribute, making them more self-sufficient than other schools and having some of the ideals of Confucianism in what was expected of the monks. Ch'an can be seen as the a form of Buddhism uniquely tailored to Chinese scholasticism, focused mostly on its own sūtras and koāns, both of which were short sayings meant to be thought of and meditated on by the individual. Ch'an's focus on personal self-improvement for the benefit of the world at as a whole fit nicely into the established worldview of the Chinese as well.

Pure Land is a salvationist doctrine based around the belief that Amitābha Buddha (not the historical Buddha, but rather a legendary godlike Buddha) would take all who wanted enlightenment to his "Pure Land" upon their death only by reciting his name ten times, this was available to everyone who did not blaspheme Buddhism or the Dharma. The basic idea easily penetrated into Chinese society and long outlived the collapse of Pure Land monasticism for the universal appeal and easy integration into the Chinese idea of heaven.

You may have heard the terms "Chinese Folk Religion" or "Vietnamese Folk Religion" at some point in your life. The idea that his refers to shamanism or animism isn't exactly wrong, but it doesn't encompass all of what it is. Rather, often this involves a syncretic mix of traditions where you will see a statue of shrine for Amitābha next to one for the heavens and the immortals or gods. The mother goddess becomes conflated with Guan-Yin (the Bodhisattva of compassion in the Chinese Buddhist pantheon) or vice-versa. What we see here is the ultimate fusion of the schools of thought into a wide variety of traditions with no clear pattern.

In summation, there is more to say about the interactions between Daoism, Confucianism, and Buddhism. Books could (and have been!) written about it. But the basic thing I'm getting at here is that throughout Chinese history, the three philosophies have been everything from complementary schools of thought to bitter enemies, yet they intermingle in a dramatic way constantly challenging one another and pushing each other forward. Beyond this, the three philosophies are fairly abstract in their scholastic form and the distinctions blur for the populace forming a syncretic mix among peasants, especially in more recent times where they are in far more contact on the local level.

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Sources:

• Robinson, R. H., Johnson, W., & Bhikkhu, T. (2005). The Buddist Religion: A Historical Introduction (4th ed., Religious Life in History). Belmont, CA: Wadsworth.

• Mollier, C. (2009). Buddhism and Taoism face to face: scripture, ritual, and iconographic exchange in medieval China. Honolulu: University of Hawaiʿi Press.

• Chen, K. K. (1973). Buddhism in China: A Historical Survey. Princeton, NJ: Princeton University Press.

They're "similar" in the sense that China, Korea, and Japan (to a lesser extent) adopted them as synchronic philosophies. But they are not similar in isolation and actually spent much of recorded history in conflict with each other.

Both emerged around the same time, ~500 years before the birth of Christ. Master Kong and Master Lao ("Confucius" is the latinization of "Kong Tzu"), as you mentioned, emerged during the Warring States period of Pre-Dynastic China. Siddhartha Gautama, the historical Buddha, was born and his philosophy originally flourished in northern India. Considering each in isolation, Confucianism and Daoism are more similar because they share a language, while Buddhism was taught and propagated originally in an entirely different language family.

Buddhism's original texts are the Tripitaka, a long series of texts written in Pali and later Sanskrit that are the reported sermons of the Buddha. Kong Tzu was reportedly a master of Yi Jing study and had some part in the composition of its commentary (though most historians seem to accept that these are written by his students, as were the Analects usually attributed to the Master himself). Lao Tzu's historicity is famously questioned, with his famous Dao de Jing being attributed to a later Daoist Chuang Tzu.

During their lives, Lao Tzu (the most mysterious of the three) is unique in that he did not seem to try spreading his own philosophy. Confucius worked as a government bureaucrat in different parts of the Warring States trying to get this king or that to follow the ways of heaven. The Buddha wandered similarly through India, though he was not as immersed in the world as Kong Tzu was. The Buddha insisted that reality was realized by joining the Sangha, the ordained community of monks, and rejecting objects of attachment and desire (i.e. family, politics, etc.). Kong Tzu, through his political efforts, insisted that proper establishment of worldly relationships.

As far as their philosophies go, Buddhism is drawing from an anti-Vedic tradition in pre-Classical India, emphasizing qualities of enlightenment brought on by the rejection of personal desires and attachments (Dukkha) through the Eightfold Path (Magga). Confucianism is built around the establishment of proper relationships, basically emphasizing that respecting a hierarchical order of reciprocal relationships would bring about peace. The Five Relationships are ruler to subject, father to son, husband to wife, elder to younger, and friend to friend. Daoism's main goal (if religions can be said to have one) is to achieve immortality, leading to a rich history of Daoist literature pointing to sexual practices, numerology, alchemy, and meditation aimed towards that goal. Though this is difficult to talk about in broad strokes with any accuracy.

Daoists and Confucians have typically been in conflict. The Han Dynasty was considered a golden age of China, but during its history, they went through periods of both Daoist and Confucian extremism (yes, that was a thing) leading to chaos and violence. During the Han period (221 BCE - 206 CE) the Silk Road became established and China's commercial interaction with India began, starting China's interest in Buddhism which was mainly academic at first. Buddhism's emphasis on asceticism, reincarnation, and extinguishing the self through the abandonment of worldly attachments, was a radical notion in China where family lineage, and subservience to Heaven was taken as a given. It took a long while for Buddhism to become acceptable in China, with a lot of suppression, massacres, and synchronization following.

In religion, we associate religion with the supernatural. Buddhism acknowledges the presence (or at the very least, the possibility) of deities and other supernatural beings. At the same time, it acknowledges that these are superfluous to the main philosophy. Buddhism is sometimes called "Atheistic" or "Agnostic," but it's more accurate to say that Buddhism is "Apatheistic." Buddhism does not care how the universe originated because that answer doesn't solve any of our current problems (i.e. that Dukkha from earlier). Confucianism and Daoism mention "Heaven" a lot, which in Chinese philosophy could be compared easily with a monotheistic "God." Since this isn't quite my area of expertise, I'll shy away from making specifics. But my experience dealing with this area of the world draws some similarities between the way that Buddhism deals with the divine as Confucianism and Daoism do (and this may be why from our Western Monotheistic standpoint, they seem similar). Chinese philosophy deals a lot with this vague "Heaven," which seems like a monotheistic entity, but seems more like a super-personal force trying to exert its will upon the Middle Kingdom (i.e. China). The Emperor of China is (was) regarded as the "Son of Heaven" and in Confucian thought, the Son should be subservient to the Father. (And if we want to extend the metaphor even further, Korea was thought of and discussed as the "Little Brother" of China, and was expected to behave similarly.) But as the sovereign over a Daoist, Buddhist, and Confucian empire, the Son of Heaven was also expected to engage in Buddhist and Daoist ceremonies and throughout history has had advisors of various power and strength compete with each other for Imperial influence and benefit toward their own schools of thought.

Long story short: They are not similar in isolation, but a shared history (mainly in China and Korea) has produced a long, shared history, and a library's worth of synchronic philosophy to sift through.

Sources:

China: A History by John Keay

Fathoming the Cosmos and Ordering the Universe by Richard J. Smith

The I Ching: A Biography by Richard J. Smith (a shortened and easier version of the above)

Korean Philosophy: Its Tradition and Modern Transformation

What the Buddha Taught by Walpola Rahula

A Concise History of Buddhism by Andrew Skilton