As someone who gets styes occasionally myself, I can tell you it isn’t a stye. They happen right at the base of an eyelash. And, my daughter had a clogged tear duct that required surgery, and it didn’t look like that, either. It was in the inner eye corner, where the tears come from. Her eye constantly watered and the whole eye was swollen.
I would guess some sort of blister that is infected and would get him to a vet. If you can’t get him in today, I think warm compresses are usually recommended. Since it drained some from the compress you used before, it seems like that is helping.
Take him to a vet, though. It likely won’t go away on its own, and could become serious and is likely uncomfortable.