This is far more rooted in culture than history, i.e. the common south Asian culture shared in the Dharmic religions (what we can loosely group together into Hinduism, Buddhism, and Jainism) is IMO more important to understanding what actually distinguishes the religious experience (and religious identity) far more than any one historical process or date that we can then point to.

There are a series of prayers and initiations that takes place in Abrahamic faiths in order to be inducted into the religion. In Christianity (while it varies between sects) you must believe in your heart and confess with your mouth that Jesus is Lord (Romans 10:9). In Islam you must perform the Shahada to declare your faith and be officially converted. In Judaism, (at least for Gentiles) you must be rejected three times by a Rabbi before you can be inducted into the religion.

Dharmic religions tend to be much more focused around commitment, devotion, and access to the guru rather than ovational inclusion in a community. What I mean is that to be officially considered a Buddhist, you must "take refuge" in a specific Buddhist teacher, declaring him (or her, but usually him) your "root teacher" (in Tibetan rtsa ba bla ma) who is your primary teacher. This doesn't necessarily mean you are an ordained monk (though that is possible) but you can take a series of lesser vows and be inducted as a lay follower. Buddhism, as a result, has four types of participants: bhikku, bhikshuni, upasaka, and upasika. In English: monk, nun, male lay follower, female lay follower.

I'm not an expert in Jain traditions, so I won't be going over the specifics since I can't elaborate on them. Suffice to say that Jainism comes from the same tradition (and same time period) as Buddhism.

Hinduism (or more accurately, the collection of religious traditions we can group together into something we call "Hinduism") works in a similar way, but throughout Indian history - basically up until the Buddha and Mahavira showed up, with some exceptions - the authority of the Vedas was assumed, and for the sake of this post, we can say that Hinduism was assumed. Aside from the fact that there was no modern science or significant outside force to threaten the religious make up of the subcontinent until the arrival of Islam (and Buddhism, but that was internal and of a rather different sort) there were several factors that lead to the reinforcement of assumed Hinduism: 1. the caste system which benefitted priests and warrior-kings and enforced social order through the systems of governance, and 2. the philosophical underpinnings of Dharma which the system is based on were self-reinforcing.

Dharma has many translations, usually used right now to refer specifically to "religion," but it is notable (especially when pluralized in Tibetan, i.e. chos rnams = "Dharmas") that it can refer to "phenomena" or "natural law." The Buddha and the gurus are not thought of as having created a religion or even revealed the path to god (such as in Abrahamic religions) but rather that they have merely uncovered something that was already there. Or we can say, is already assumed to be there.

Even the foundational texts of Indian religion, the Vedas, are uncreated and assumed to have no originator except god himself. The Upanishads, which were later conducted by gurus and their followers serving as to explain the Vedas, came much later, and are explicatory of this foundational law.

"Hinduism" is not so much as a single religion as it is a complex of religions with their roots based in the same foundational texts. Vaishnaivism, Shaktism, Saivism, and all the other small branches of Hinduism all assume the divinity and authority of the Vedas, but disagree on which avatar of god predominates, and which section of the Hindu corpus of work. I'd like to assume that (like Buddhism) Hindu gurus pass on their lineage to their students (who ideally should be their sons, as Brahmins, the highest Hindu caste, are to pass their teaching down hereditarilly, in practice I think this is very much varied) who then pass teachings on beyond them. We have many texts from Indian traditions that are no longer taught. At some point in history the texts just fell by the wayside (either a guru didn't teach it to a student, or the one student who did learn it passed away, or a teacher decided to refute it, etc.) and that particular tradition encased in a text now serving as an academic study, is no longer a living document passed down through tradition. Whereas others, though they may be static in form, are very much alive (i.e. the Bhagavad Gita to take the most cliche of examples). The lineage of accepted traditions, rituals, and texts (both oral and written, usually some combination) are what make an Indian tradition a part of the tradition, and following a guru in that tradition is what declares you as a part of that lineage. If your teacher is a Saivite, he will focus on Saivite teachings, thus making you Saivite.

The same is true in Buddhism. When you follow a Buddhist teacher, i.e. take refuge in one, that makes you a Buddhist. And as Buddhism began to split (intentionally or otherwise) declaring yourself for a Buddhist teacher makes you a part of their tradition. What united Buddhists - though the Sangha very quickly split and spread and formed new collections of Buddhists - was their faith in the teachings of the Buddha and their rejection of the Vedas. But as far as the identity of what made one "Buddhist" or "Hindu" is a matter of their teachers.

Simply put: if you followed a teacher who rejected the authority of the Vedas, that made you a Buddhist. If you followed a teacher who preached the authority of the Vedas, that made you a Hindu. If you were not a declared follower of either, then it was probably assumed based on the place you were born and what kind of initiation rituals are performed at your birth.

This doesn't necessarily mean that people went around calling themselves and declaring themselves Buddhists, or even that Buddhists went into Hindus faces denouncing the Vedas (though this did lead to some very intense debates, see Asvaghosha). In fact, I'd argue that Indians were more ready to declare themselves as a part of a lineage rather than identify themselves as "Buddhist" or "Hindu" which seems to me to be much more of an imposed Western concept of religious identity.

In The Myth of the Holy Cow by J.N. Jha, he traces the origin of the sacralization of cows in Indian religion and thought (and the ridiculousness of the Idea that it comes from the Vedas) pointing out that cow protection is an identifying mark of Hindus vs. Muslims stemming from the 9th century. A millenia later, when Indians were forming more collective identities under thinkers like Dayananda and Tilak, they began emphasizing rituals like Shivaji, or Ganesh Chaturthi to promote a "Hindu identity." This movement, called Hindutva, emphasizes the notion that India is a "fatherland" and a "holy land" forming an umbrella of Dharmic religions (Hinduism, Jainism, Buddhism, and Sikhism) which otherizes Islam and Christianity (historically, the religions of invaders on the subcontinent). This is still a hotly debated issue.

Early Buddhists were clearly distinct from early Hindus (though they at the time would probably scoff at the term "Hindu" to begin with) in that the former rejected the authority of the Vedas and the latter did not (and from there the philosophical gap widens). Though at what point they identified themselves under certain umbrella terms is less clear and will vary with time, location, and culture. Especially since the more ready identification in Dharmic religions is with one's guru, not necessarily with one's religious group.

It is a fact that Hinduism and Buddhism have never been and still are not separate from each other in South Asia

Can you elaborate on what you mean by this?

Firstly, Hinduism is not even an Indian term. Today, the religion that is identified as Hinduism in India comes form the school of Brahmanism. Although that too would not be an unfair assessment. One could say that the Brahmanistic tradition is the dominant one amongst the many that together were referred to as Hinduism by European colonialists. These colonialists in turn were guided by the ancient greco-roman records. The greeks first referred to Indians as Indu - settlers on the river Indus.

Back then India was a boiling pot for a number of schools of thought (just like ancient Greece and ancient China). The ascetic tradition was quite common in India. Some of the acetics were driven by the alure of supernatural attainments. Some remained solitary, while others had a large following. Some were true renuncees while others were householders.

Over the years, as the Aryans exerted there dominance over the region, Brahmanism (the school backed by the Aryans) gained popularity. As the name suggests Brahmanism was centered around the deity called Brahma. Even so other local traditions managed to survive. Like the nagas that worshipped Shiva. The worshippers of the tiger goddess. Those who worshiped Indra. Sun god worshippers, wind god worshippers etc. Some non-thestic schools managed to survive as well.

Even though the Arabs used the term hindus first, to refer to Indians. It was the European colonialists who clubbed all the insitu theistic religions into a single group, making Hinduism the religion that it is know as to the world and to the Indians today. That is why you still find so much variety in the religious customs and practices of the 'hindus' all over India.

Buddhism is completely different from Hinduism. During it's hay days in India it served as a strong opponent to Brahmanism. The fall of Buddhism can be attributed to the disciples of Buddha who though diligent at preserving their master's teaching were not as adept as the Buddha himself at taking the teaching to the masses. And to the Muslim (Turk, Persian, Arabic) invasion of India.