Strictly speaking, copyright applies to a particular expression of an idea — in this case, a recording of a performance. Copyright also applies to the arrangement — you can’t (legally) sell copied sheet music.

As a practical matter, artists seek permission to cover (hah) their bases and because the recording industry is functioning like a cartel. Don’t cover our songs and we won’t cover yours without permission.

But we saw Taylor Swift cover her own songs. She didn’t seek permission (or did and it wasn’t given). She now owns the copyright to the covers, but not the originals. The owner of the originals is mad about it but she doesn’t care about maintaining a good relationship with the owner.

She might be on firmer legal ground since she wrote the songs and was the original recording artist, and will win in the court of public opinion, but labels hate that idea and will fight it anyway.

But you see why artists are much more prone to cover others’ music during live shows. They have a stronger legal defense since it isn’t a recording and would ruffle fewer record label execs’ feathers.