Yeah, it definitely sounds like they're going for the "my wife's boyfriend" meme.

They could just say "son" too, but that might depend more on their relationship.

Stepson might involve some kind of adoption, or, I don’t know… My dad remarried when I was in my late teens, but I didn’t see him much. His wife has never been referred to as my stepmom, etc, but my situation might be different due to a lack of closeness. The OP clearly has some kind of relationship with the kid since they are going to pick them up, but I can see how they might not have that kind of label.