I can't say with authority that India is the only country that ties vaccination record to passport, but it's certainly going to be an outlier. I don't think we have any other reports of this happening, so I suspect actual experience is going to be limited.
I'm also not clear what exactly an Indian "travel certificate" is compared to what most of us would expect to see on a vaccination record, so you may need to expand on that to get better advice.
I would still only expect a check-in agent to be interested in your name and the vaccination record, not the tie-up with the passport, but I agree it would be sensible to carry both.
You do have another option of producing the result of a PCR test within 72 hours of your first flight if you want a back-up. This potentially also subjects you to a LFT test on arrival.