It's already happening, they aren't the best of pets but selective breeding for personality traits that make for good pets yields results even in 3-4 generations worth of breeding it.
A lot of animals
can be domesticated without issue, it's up to the owners to understand how to provide for them and communicate.
This Lady is a solid teacher and isn't as gimmicky as some of the "Fluentpet" types that became a MLM scam. Bird Kindergarten has scientist-backed research behind them.
This lady has been slowly introducing a wild squirrel to the same activities she does with her birds, obviously modified for a squirrel. Again, completely wild but still picking up and learning skills she has taught her birds.