Unlike Christianity and Islam, Hinduism does not depend upon historical events, geographical places, or even people for its context to be understood. Atman is not Indian. You don't need to be Indian to practice Yoga, and nowadays you don't even need to be a Sanskrit speaker to read and understand the gist of the scriptures. The core of Hinduism is not concerned with any particular person or event in the distant past and one's particular belief about it, but rather is concerned with the nature of oneself here and now. Whether or not one is Indian physically.
I think your opinion viz. Hinduism cannot be distinguished from India is very odd especially when you are contrasting it with Islam. To my eyes the opposite is true. Islam is an Semitic religion. It's an arabic adaption (or retcon) of judaeo-christian theology. Muslims pray towards Mecca, which is in Saudi Arabia. Muslims revere Muhammad- a historical personage from Saudi Arabia. The hadith are concerned with the events of his life in Saudi Arabia and his teachings given to the Arabs he was converting to Islam. The Qur'an is a 'miracle' of Arabic literature, and Muslims say one must learn Arabic because a translation of the text is no longer considered revelation. The concepts of paradise as an oasis and polygamy within Islam reflect cultural and social values of the desert-dwelling Arabs in seventh century Arabia. The imposition of sharia law is arguably the imposition of tribal Arabic modes of government.
The Qur'an, as the Bible, is deeply concerned with the activities of a few people living in distant Israel; its context is local and historically orientated. If you take away the historical events and personages, there is nothing left in these texts. The same simply can't be said of Hinduism. Okay, one might argue that the Mahabharata seems to be making claims that Krishna was a historical personality. Let it be so - whether Krishna was or was not a historical personality, whether or not the great Mahabharata war ever took place thousands of years ago, in no way affects or refutes the validity of Krishna's words in the Gita, or the philosophy and spiritual teachings within the epic. Whether or not the Upanishadic sages, named in the Upanishads, really existed is irrelevant, because the teachings therein stand entirely on their own merit. They don't need a historical context or a personality attached for them to be true. They apply to all people in all times, even now. We don't need to have faith that any of these people existed. It's like saying the Buddha didn't exist. If there was no historical Siddhartha Gautama, then nothing would change, because the teachings are still there. And if the teachings are still there, then we should attribute them to somebody. Let's call that somebody the Buddha. Nothing changes. And likewise, Atman is still Brahman, whether Yajnavalkya said it or not. Whether Uddalaka said it or not. Whether Krishna said it or not. It's universal. So I strongly disagree with you here.