When we talk about Hinduism, we are talking about a very broad group of traditions, broader even than when we say Abrahamic Faiths. Hinduism, as it stands now, incorporates thousands of individual groups many with their own distinct philosophies and worldviews that are counter to another. Most Hindus recognize the authority of the Vedas and the Upaniṣads, but not all. In any case, it is hard to talk about Hinduism as a monolithic tradition, but I will do my best.
When we talk about the Influence of Monotheistic faiths, we are typically referring to Abrahamic, due to our Western bias, but what is a "monotheistic faith" in the first place. Islam decries Christianity as Shirk (idolatrous or polytheist) due to the doctrine of the trinity or worship of saints. However many Muslims also venerate saints or ward of spirits in a way that is difficult to distinguish from what they call shirk. We largely all agree that the Greeks were polytheists, but many believed in an all powerful personal god personified in a number of forms, such as the powerful Platonists of late antiquity. Bearing this in mind, now we can look at Hinduism.
Hinduism is very diverse, but there are large sects and strong themes throughout. One of these is the concept of Brahman, a universal monistic principal that pervades all things. Many Hindus identify this principal with a God or entity, such as Śiva or Viṣṇu. In essence, this principal is the form of a supreme god pervading all things while the human forms and stories are merely personifications of the God. This Supreme entity varies and often exists alongside other gods, but does this make it not "monotheist?" The gods of Hindu, especially in its Bhakti incarnation, are personal, all-powerful, and all pervading, like the God of Christianity, the God of Judaism, or the God of Islam; so is it not a Monotheistic religion? Of course, not all Hindus share this view of God or gods, but I would say that the majority do.
IN any case, philosophy aside, how DID Hinduism survive in India, while it did not elsewhere (such as Indonesia and the Philippines) and many other faiths didn't either. I think we need to move away from Monotheism as the determining factor here, after all, Zoroastrianism is almost extinct and it was Monotheist in the classical sense. Rather, we need to look at why people "convert" and how religions develop and shift.
South Asia circa the tenth century was perhaps the most religiously diverse region in the world, rivaled perhaps only by China and Persia, but the later had recently seen a dramatic shift in religious policy as Islam came. That isn't to say that Islam destroyed religious diversity though. There is a persistent myth that Islam was intolerant and religions were swiftly crushed, however, Zoroastrianism, Nestorianism, Buddhism, and other faiths continued to an extent under Islamic rule, though some suffered more than others. In any case, the tenth century marked the incursions of Islamic Turkish invaders into India and the formation of the first Indian Sultanates. The religious policies of these Sultanates varied, with some being very active in persecuting other religions, especially in the west, but others being very open. Furthermore, early on there was an encouragement of dialogue between Islam and Hinduism. The Ghaznavid's encouraged this dialogue and facilitated it through scholars and polymaths, most famously Al-Biruni. Even the (somewhat questionably and unfairly) iconoclastic Dehli Sultanates often encouraged dialogue, with some like Sultana Raziya, even funding Hindu temples.
There was also an economic incentive to keep Hinduism strong in the empires that ruled India. Hindus, Jains, and Buddhists had to pay the Jizya tax (and sometimes others) in order to practice while Muslims did not. This meant that every convert, while perhaps being more reliable, payed less into the tax pool. In essence, this meant that keeping the Hindus around made sense and made dynasties very rich. We also have to keep in mind that while large parts of India were controlled by Muslims, many local aristocrats and powerful land owners were Hindus who swore fealty. Most of the Persianate dynasties had a somewhat decentralized powerstructure, culminating in the Mughals whose system eventually meant that these rulers were only nominally under their control after the rule of Aurangzeb.
The Mughals are very interesting as, perhaps more than any dynasty before them, they took advantage of the existing rulers and promoted religious tolerance by having a number of Hindus and Jains in fairly powerful positions. Akbar, the third emperor, even suspended the Jizya tax and promoted his own synthesis of Hinduism and Islam, under the guidance of radical Sufis. While his legacy of religious synthesis was not adopted by his successors, his policies of keeping Hindus and Jains in court were.
When the British took control of India, they inherited the systems already in place. The British were largely unconcerned with converting the populace of India, or at least, the ones with the most power weren't. There was of course a faction that did their damndest to convert the enormous population of India, but they were not particularly successful, outside of the region of northeast India. The British also held a sort of "divide and rule" policy which encouraged the tension between religious groups to make them more submissive to the British. Furthermore, the Romantics and their successors found themselves drawn to Hindu ideals and religion, with translation efforts skyrocketing in the mid nineteenth-century.
Sources