There are actually two copyrights and sets of rights in play here: one (the © copyright[1] and the "publishing rights") is on the song itself, in the abstract - the melody and lyrics. This is owned by the songwriter, and usually assigned to a publishing company.

The other, (the ℗ copyright[2], and "master rights") is on a specific recording of the song, and is usually owned by a record label.

The cover generates royalties for both the recording artist (or their record label) that recorded the cover through the ℗ copyright and the songwriter that wrote the song (or their publishing company) through the © copyright.

1: https://www.copyright.gov/forms/formpa.pdf

2: https://www.copyright.gov/forms/formsr.pdf