"She owns him not."

I always kind of wondered about the line in the book that says, "“his cat he calls her, but she owns him not.” I assumed this was some archaic turn of phrase, though I've never encountered similar use. Also, someone on Quora asked if it was an error. One answer says, no, Tolkien phrased it that way because cats own their person, not the other way around.

I'm highly skeptical of that answer and wanted to know if anyone can clarify.

It makes me think of the word 'disown', which means to refuse to acknowledge that something/someone belongs to or is connected to you.

So I'd read it as 'Sauron thinks of her as a pet, but Shelob does not acknowledge such ownership or connection'.