In this case the writers are talking about "publishing" rights which are distinct and separate from the sound recording (I believe those are called the "master" rights).

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Copyright is split into two main sections: copyright in the song (known as publishing rights) and copyright in the sound recording (known as master rights). The publisher only deals with the publishing right, which is the songwriting side and includes the music and lyrics.

Traditionally, a record label will own the master right, which is basically the right to use a particular recording of that song, but if you’re a self-releasing artist or producer then you will most likely own this right yourself.

https://dittomusic.com/en/blog/music-publishing-explained-fo...