I heard him say “Hindi”. It actually sounds a bit like French near the end, maybe some influence in it? Is it on the Omniglot site?

I guess Kannada

Does Kannada have a “sh” sound, though? On the page for its alphabet, I didn’t see one.

Kannada has a palato-alveolar fricative which I guess can be called a “sh” sound. Omniglot does have it listed. In some other Indian languages, it is often pronounced as a postalveolar fricative, although it traditionally is regarded by Indian grammarians as a palatal fricative.

To me, it sounded Dravidian. I think it may be a south Dravidian language like Tamil, Malayalam, or Kannada rather than central-south like Telugu. I was leaning towards Malayalam. But I do not have enough exposure to Dravidian languages to be sure.

The answer is Malayalam.