I can only guess but I love finding these things out, can a horse get an abscess as well as humans can (like a tooth/jaw abscess)? And are there any certain things for sure that a horse can not get that a human can?

Horses can get them too. Tooth root abscesses are fairly common, and require some serious head surgery to remove the tooth and let it drain. If you ever see it you'd almost mistake it for brain surgery befauce the vet had to get into the huge sinuses of the skull.

Lymph nodes behind the jaw also abscess frequently in Strangles, and tey can get skin abscesses after trauma too.

Since horse teeth grow continuously it's a little harder to treat an abcess than with humans. You can't just pull out the tooth. You have to treat the abcess with antibiotics. The biggest concern a horse can have is if the tooth cracks downward. If it splits down the middle it can be quite dangerous.