I don't think there is any single "right answer" to the question "what is a language". In fact, that's why I was wondering about your opinion, as a speaker of Hindi, as to how linguists would tend to classify it. I wouldn't want to assume that, as a linguist, we are right and Hindi speakers are wrong! That's a rather silly position to take... but then again, most speakers of a language don't necessarily study their own language. As for political bias, I think we all have that - even linguists who take the political position of having no position, which is a political position whether we like or not! IMO...
I would say, based on what I've read about it, that there was many centuries ago a language, probably only spoken, which may not have even had its own name initially, and was itself a dialect related to other languages spoken in the area which is now northwest India and Pakistan. Slowly over much time, this language has split into the dialects/languages now referred to as Hindi and Urdu. As is often the case, this splitting happens slowly, gradually, and incompletely. Perhaps in a few centuries Hindi and Urdu will no longer be mutually intelligible, and undeniably separate languages - but that hasn't happened yet (if it ever will). Complicating this process of change are issues of politics, empire building and collapsing, nation-building, ethnic identity, religion, etc. I won't pretend to know anything beyond the mere basics about any of that, though...
As far as linguists are concerned, it would not be syntax entirely but rather the totality of the *spoken* language, including syntax, vocabulary, phonology, etc. - mutually intelligibility is the criteria most often cited, as this would cover all of that. This leads linguists to classify Cantonese and Mandarin (same script, almost no mutual intelligibility) as different but related languages, while the gov't of China takes the view that because they use the same script they are simply dialects of one language. When it comes to Hindi and Urdu, it seems from your description that an "unbiased" linguist would have to classify them as very close dialects of the same language, since the average Hindi and Urdu speaker would have little trouble speaking to each other. Perhaps much less than speakers of certain dialects of Italian, or German, or certainly Chinese.