There is nothing revealed in either the show or the movie about what happens to Max after the Von Trapp escape. Max has the personality of a survivor who can adjust to circumstances(This is clear in the film version, but clearer in the stage version with "No Way To Stop It".) If you want to speculate, I would guess he survives the war. If I have a chance, I'll check Maria's autobiography sometime to see if it says. I haven't looked at it in a long time. It's the book that the show was (loosely) based on.

Max is not a relation. He is a family friend. It's true that the children call him Uncle Max, but that's just as a traditional sign of respect that was often extended to the closest friends of the family.