Yes, this is something dealt with both in history and sociology, the process is known as Sanskritization.

First thing to know is that unlike religions like Christianity, Hinduism is extremely diverse in nature. It has no singular holy book (though the Vedas are acknowledged by most groups), no single holy prophet etc. As a religion, it can function both as polytheistic and as monotheistic as the Vedanta and Upanishad scriptures contain distinctly monotheistic ideas.

However, in the vast majority of cases, Hinduism is polytheistic, and it is this nature that let it absorb deities of other cultures. New deities could simply be added onto the pantheon and this was a very local process, that is, each region would have its own localized deities. Orthodox Brahmanical Hinduism was mostly predominant from the Punjab area in the Northwest through the Gangetic Valley of Uttar Pradesh till Bengal in the East and then in several parts of South India. As Hinduism expanded it came into contact with the numerous tribal cultures which inhabited the forested hills and plateaus Central India.

For the next part of my answer, I will be specifically talking about the situation in Bengal. In the medieval period, more specifically in the 14th - 18th century period we find a particular subgenre of folk poetry called "Mangal kavyas" Each Kavya or poem focuses on one specific deity, which is always a deity hitherto worshiped by people living on the borders or outside the influence of Hinduism and how that deity came to be worshipped by the protagonists who are always Hindus. Some famous Mangal Kavyas are Manasa Mangal and Chandi Mangal. The narrative is typically that the deity offered favour to someone in difficulty and they overcame many troubles in their life and established their own kingdom/got rich and established the worship of that deity in society (chandimangal) or the deity confronted a respected high caste member of Hindu society and forced him to worship. In the Manasa Mangal, the high caste merchant protagonist loses his family to snakebite, and his fortune before finally submitting and begrudgingly worshiping Manasa (the goddess of snakes)

It has been theorized by historians that these Mangal Kavyas reflect an ongoing socio-cultural phenomenon where the tribal deities were being absorbed into ortohodox Hindu society by giving them a place in the pantheon. Kavyas like the Manasa mangal also seem to hint that this was not always a smooth process. Very importantly, in Manasa Mangal, the person who is forced to worship Manasa was originally a devotee of Shiva, who is one of the chief gods of the Hindu pantheon. This person clearly holds Manasa in contempt, going by the lines ascribed to him, but his act of worship is all Manasa needs to gain social acceptance.

The best known work about this genre of poetry and its historical importance is by David Curley, this link should give you a look at the table of contents of his book, as far as I remember he also has journal articles on this. However please note that this is still an active area of research, and new discoveries in the area of folklore history can greatly expand our understanding.

There is one more thing I wanted to say - religious identities in many areas of India in pre-colonial times were fuzzy. A good example is much of East and South Bengal. It has been shown that many of the people who lived here, who were either peasants or foresters by profession had complex, syncretic religious identity. Their poetry and song refer to Islamic, Hindu and local folk culture ideas. Their patterns of worship would touch on folk deities like Manasa, tiger deities who controlled the notorious Sunderban jungles, Hindu deities like Krishna, and Islamic Sufi pirs. There is a morning hymn that salutes the sun and the Himalayas (deities in Hinduism) and then Mecca and Mohammed in the next line. Religious identity was often thus constituted out of many different themes and not traceable to any one tradition. One of the best works on this aspect of history is by Richard M Eaton: The Rise of Islam and the Bengal Frontier, 1204-1760 - Oxford University Press : 1993.

I hope I have been able to answer some of your questions, and that this answer is upto the standards of this subreddit.