Time to fess up.
Confess showed up in the late fourteenth century (shortened as “fess” in 1840), coming from the Old French confesser, Vulgar Latin confessare, and classical Latin confess-. Now that’s from the verb form of confiteri, to acknowledge, a mix of the prefix com- (together) and fateri, to admit. So it’s “to admit together”? Fateri can also be traced back to the Proto Indo European bha-, to speak or say, which is a part of, like, a bunch of words. So many.
The other fess word is profess, which showed up in the early fourteenth century meaning to take a religious vow. Obviously it’s related to profession, but will get into that later. It’s also related to the Old French profess and before that the Medieval Latin professus, avowed. It’s related to the classical Latin profiteri, which could mean volunteering, profess, profession, acknowledge, or make a public statement of. The pro- is from per-, meaning forth, and with fateri, it would be something like “to admit forth”. Which makes sense for the public statement thing, but not so much the other definitions.
Profession is even older than profess, having shown up in the thirteenth century. Although back then, it meant the “vows taken upon entering a religious order”, coming from the Old French profession and classical Latin professionem. It didn’t mean someone’s occupation until the fifteenth century in the sense of an “occupation one professes to be skilled in”. Saying you have a profession is publically claiming you have skill in something. And professor showed up in the late fourteenth century meaning a teacher of a branch of knowledge, coming from the Old French professeur and classical Latin professor (leave it to French to add extra letters). The etymology dictionary doesn’t specifically say why professor became another word for teacher, but I’d guess that a professor is someone who is professing so much skill at something that they can teach it. Kind of ironic since the one thing I’ve found professors to suck at is teaching.