So about a month ago, this red growth appeared. Just this week, it's doubled in size and a faint ring appeared around it.

The total diameter  is just under an inch.

I've taken him to the vet thinking it was skin cancer, but he told me that's it's probably a wart or something. Definitely  not skin cancer and to not worry about it.

He's a 4  year old pitbull with really bad skin allergies (which is said why the hair loss).  He was born in Georgia, so I'm sure he's allergic to something out here in AZ. He's on allergy medicine and is actively taking allergy pills/ monthly shot.

The growth is located on the right side of his chest

Are there a lot of ticks out there? a "bulls eye" spot is typically associated with Lyme disease. Otherwise I would say maybe it got into something else it is allergic to. My Cane Corso gets them sometimes if she gets into certain weeds or seafood.

My Boston Terrier has one also. We sprayed it with GenOne Spray from the vet twice a day. It went down a lot but starred to come back when we got lazy spraying her. The vet then gave us antibiotics and it helped but it's persistent. and we are still spraying her twice a day.

My RR had one on his face by the eye last year.  Vet diagnosed it via a microscopic skin cell exam (take sample on a slide and examine via scope).  Cutaneous Histiocytoma.  Normally resolves without treatment in about 3 months.  It cleared up just like the vet said it would.

Here is a useful page on cutaneous histiocytomas. LINK