It seems pretty clear that the Fulton County Sheriff’s office holds the copyright here; the only real question is in what constitutes fair use of a mug shot.
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.
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.
In the world of post-truth drama, it makes sense. Nothing turns around the loyalty he builds when he captures the hearts and minds of his voters. Even if he dies of a heart attack, people will still seek out the kind of attitude he has been iconic for. Eton Musk is 25 years younger, but seems to be following Trump as a behavior model of successful businessman icon.
God, it would be hilarious if Trump was forced to pay Fulton County a big lump sum and destroy all of his remaining merch. I doubt the county has the resources to wage that kind of legal battle, but one can hope.