Where I have to wonder if for a cat, I could train them to have a sound I play on their phone to be their name for all they care (kinda going back to them possibly thinking a ringtone could be my name). It's still just a sound that they are responding to not a word.

Maybe like you said though, the difference doesn't matter. Maybe even as humans while we understand that a name is language it is still something else entirely? I don't know, that is a realm of cognitive study I know nothing about. But it is fascinating.