We don't know.
Certainly Lore treats him as such as soon as she sees his concentration camp tattoos, regardless of his fake papers (which she later learns about). This explains her behaviour at the end of the film. Whilst he was using a dead Jew's papers, it is a jump to go from learning his papers are fake to believing everything about him is a lie. Using fake papers would still have been easier than having no papers at all, which seems to be the only alternative.
He is a survivor - both of the camps and beyond. Once leaving, and without any papers, he continued to adapt, doing what he had to to survive - including taking the dead man's papers.
He does tell the young boy that he isn't Jewish, but it could be argued this was to avoid scarring the boy anymore - the ideas of Nazism he had were his parents, not his.
This also brings a nice full circle to the story, with Lore ending up trusting the person she was taught to hate. Thus in the film there's a strong logical argument that he is Jewish.
The director, Cate Shortland, revealed that even Thomas' tattoos were fake:
Interviewer: [Thomas] could have been an SS man himself for all we know.
Cate: He tattooed himself. That tattoo is not from Auschwitz. I loved that –
that everything is slipping through your fingers, in Germany in 1945.
Nothing was real.
Knowing that changes things. If those tattoos aren't real and his papers aren't real, then suddenly things seem very different. To me, it suggests he was an SS guard. That would give him intimate knowledge of the concentration camp tattoos, knowledge of the camps and access to papers and knowledge that posing as a Jew would have been less dangerous than in his real guise.
It also presents a wonderfully wicked irony to the story - Lore learns to question and disagree with her parent's bigotry through a man she believed to be Jewish, who was in fact everything her parent's bigotry represented.
Consider the meaning of the word lore: a body of traditions and knowledge on a subject or held by a particular group, typically passed from person to person by word of mouth.
The main focus of the film is Lore's changing beliefs about the world around her, from her parent's authoritarian and narrow minded view, to one of her own. "Thomas", whoever he is, proves to be a massive part of that.
His true identity is a mystery. But the fact he helped open up Lore's mind (either as a Jew, a regretful former guard or a guilt free former guard) isn't in doubt - and that's the message the film maker wanted to leave us with. Don't question his background - question her transition from one way of thinking to another.