How can I disown the cat?

You can't. The pet (either a dog or a cat) is a permanent addition to your farm. They're also (minor spoilers about an event late into the game)

a criteria for the end of year 2 evaluation, where if you've been patting it enough during the time you've had it for it to have 4/5 hearts with you, it counts as a point towards your rating. Taken from here.

Keep in mind that the pet is optional; when Marnie offers it to you you don't need to keep it. You could probably look up how to edit your save file if you really want to get rid of it.

Although you can't officially get rid of your cat, it's worth keeping in mind that like any other creature or person, if you keep walking into them, after a few seconds they'll start to shake slightly and then you'll walk right through them. That will allow you to get past when the cat (or anyone else, for that matter) is in the way.