I’d say a lot in music is about structure, patterns, repetition etc. And endings of words being similar is part of that.

Humans love recognizable patterns and symmetry. Rhymes sound alike, and when coupled with meter, help make the words “stick” in our memory. We also like artificial things that are easily distinguishable from the chaos and entropy of the natural world. We enjoy our feeble attempts at imposing order.

I like the novelty/predictability ratio idea. There is also the idea of “create expectations and satisfy them”, which leads to a sense of stability. Our cultures and genres create expectations. Rhymes tied to a certain metric can become part of these expectations. Of course, you can also create expectations and frustrate them, which leads to a sense of instability. Searching for “fakeout rhyme” videos makes this evident. Pat Pattison, an expert in songwriting, could be a good source on this ☺️

Pattern seeking brain

Because it’s drugs for your brain. Your brain expects to find a pattern and rewards you when you do. Brainscans have shown similar activation patterns in people that are listening to their favorite music as people who have recently injected heroin.

Rhythm

I want to add to the question with: and why does sometimes rhymes or phrases in other languages sound even more pleasing? “Sometimes friends” in Japanese is just super fun. Tokidoki tomodachi!

I have two theories, applying not just to rhyme but to traditional verse forms in general (i.e., formal constraints like rhyme, meter, alliteration, etc.):

1. In prehistory—when all knowledge was transmitted orally—verse constraints acted as a sort of verbal checksum to prevent transmitted knowledge from getting corrupted accidentally. And the presence of verse patterns became a subliminal flag indicating that whatever was being sung or recited was important knowledge worth the extra effort of casting into verse.

2. It’s been found in many different contexts that humans are most drawn to information with a novelty-to-predictability ratio of about 20–25%: if it’s much less than that we get bored, and if it’s much more than that we get lost and/or dismiss it as gibberish. So adding a predictable element like a regular rhyme pattern gives the creator freedom to add more novel elements without losing the audience.