My guess is that he isn't. It's just a young boy that reminds him of either his actual son, or, maybe just himself as a young child.
The first theory is just the general unlikelyhood of meeting up with his long lost son if he really was the offspring of another woman from an affair years earlier.
The second is bolstered by maybe thinking of the ending as a projection of Saul himself as a young boy, who maybe squeeled on neighbors, either by intent, or by sad accident. He projects the young boy he sees in the camp as a version of himself and wants to put the past - HIS past - to rest in the most holy way possible considering the dire circumstances. Thoughts???
I think the boy at the end was just a random child and not the boy back at the gas chamber, which most likely wasn't Saul's son at all given the fact that at one point his fellow worker tells him "You don't have a son." I think the initial boy, as others have stated, just connected with Saul as the result of his survival and he sort of "adopted" him from then on to give him a proper burial.
All that said, I didn't find the film to be as effective as other films on the subject.
I don't think the boy at the end, or the boy Saul was trying to bury were his actual son. I think the boy Saul was attempting to bury might have been a representation of his son, who possibly was killed before this film ever took place. I was admittedly a little bit lost with this film, but from the beginning I felt like Saul was looking for something, but nothing was really too clear.
I think something clicked in him that saw the burial of the boy as a way to somehow redeem a part of himself, in a way he possibly felt he had failed his actual son. I obviously could be wrong.
I think the boy at the end is simply a representation of life. The young boy is free and alive out in the wild, and with a chance, although slim as it may have been at that time, to be free. I think Saul saw that as a sign from God and finally felt a shred of peace seeing that boy.
The thing that confused me was the soldiers at the end. The one soldier took the boy and held his mouth shut, then let him go and he ran off and we hear gun shots. Were the men in the barn found and killed. Was the a German boy? He had blonde hair if I remember correct.
To me, the boy he desperately tries to bury is the projection of his inner self. One piece of humanity that left in him. Basically he is a dead-man walking. He marches people to the gas chamber to be kill, he cleans the corpse, his morals and values are destroy. At least by burying the boy, he can express the last bit of humanity that is left in him. By burying the boy, he saves himself. As the burial also symbolize his own burial.
The last scene when he smiles. I think he kinda has a delusion about the dead boy's soul comes and say thank you or goodbye to him. (Of course, the boy is real.)
No. It is remarked in the film that Saul does not have a son. It was symbolic of Saul's madness and clinging to his humanity and Judaism to see the boy as his son who must have a proper burial.