Taking your description at face value, something is incongruous between how you present yourself in person vs. on your resumé. I can't see how otherwise despite your qualifications on paper and in your phone conversation why a company would spend their time bringing you onsite for a specific position, and then find you unfit but still want to consider you for something else. There seems to be a disconnect that doesn'…
Some thoughts.. I'm Indian-bodied but was born and raised in the states. When I was in my early 20s I lived on and off in India for about 5 years (about 6 months at a time). So I've had a decent amount of exposure to Indians of all types (Indian and American born). I've also worked with a fair amount of Indians over the course of the last few years.
These may not be the case with you, but as someone who has had a fair amount of exposure to Indians form India, and as someone who has coached and interviewed a fair amount of Indians for a variety of roles, the following may help:
+A lot of it boils down to personality. I've noticed a very wide spectrum of attitudes coming from Indians. I think a fair amount of introspection and awareness about how you are coming across to the other side is a really powerful tool to employ. I've seen a lot of ego and general lack of sophistication and at times class from some Indians that I've met, and I'm certainly not saying that this may be the case with you, but it may be worth considering if there's something about your mannerism / tone that maybe rubbing people the wrong way.
+Being a FOB language skills can be a natural weakness but judging by your post this doesn't seem to be the case. I think being yourself and honing your personality will have more impact, but taking a little extra care in how you communicate can really work wonders.
+You say you're in a product management role, there's generally an axis of management skill vs. technical ability. Some companies may be looking for one or the other, and in the interview you may or may not be emphasizing the one they're looking for.
+And finally there's leadership ability, particularly important in management roles. Have you led your teams to success in your career? Reflect about what got you to that point, and try and channel that into your interviews.
+Above all, stay the course if you can. If you're convinced you're on the path you want to go, then persist. If you want something else and have the runway to pursue it, then a change of environment is a great first step towards finding your next step into something that motivates you.
No offense intended in any of the above, and best of luck to you!