I really wonder where this came from, evolutionarily or culturally, since it seems that only humans seem to have a response to rhythm. Some of the more talkative birds seem to as well.

This reminded me of something Milford Graves said in a book (paraphrasing): "The first sound we encounter is our mother's heartbeat in the womb -- and it swings!" [1][2]

[1] https://www.inventorypress.com/product/milford-graves-a-mind...

[2] https://en.wikipedia.org/wiki/Milford_Graves

IIRC rhythm perception is extremely sophisticated in terms of brain function, involving several regions working together in complex ways. It's built on the functions we use to model movement through space and has a timing-based prediction component plus of course the auditory processing. I don't know enough about it to confidently say but it seems reasonable to me that most animals simply don't have the prerequisite gear.