Why Trump could be making a big mistake by fundraising off his mug shot

Likeness is part of trademark law.
It's a whole separate thing from copyright.

A model can own their likeness, but they can't sell shirts of their photos from magazines.
Those specific images are owned by the magazine.

Same thing applies here.

They don't own the copyright on someone's likeness. They own the copyright on that specific image.

But there might be a fringe use case where they can block other parties from profiting off of it themselves. There are laws in some states prohibiting criminals from collecting proceeds from their own memoirs and other works, though that probably only applies to convicted felons, and Trump isn't there quite yet. But there might be precedent if someone really wanted to take this to court.

That's not what's being argued. You don't own a picture just because you're in it and this has been proven legally on numerous cases, the story behind the LUL Twitch emote being a relatively recent example.

Trump can take a selfie and put it on a shirt. If I take a picture of him and upload it to my site, I can sue him if he uses that picture instead. The argument is that the jail can too.