He doesn't name himself while you can still switch characters, though. It was afterward, during the Vault-Tec questionnaire that he chose his name and set his SPECIAL stats. You can edit the appearance of both at the start, but once you leave the bathroom mirror you've chosen which one of them you will play as.
EDIT: I suppose it's possible that you could also name your spouse during the Vault-Tec part, and they just didn't show it in the demo to keep things moving along. Once that part of the demo was over, everyone had a name--you could see "Talk to Nora" when the cursor hovered over the wife, then Codsworth addressed the husband by name and mentioned Shaun the baby.
EDIT 2: If you can in fact name your spouse when talking to the Vault-Tec guy, then maybe the baby isn't locked in as "Shaun" like we've assumed. Codsworth was the only one to say his name (the wife just referred to him as "him" and "the baby"), and we already know Codsworth has all the extra voice recording for popular names. Hmmm...
EDIT 3: Scratch what I said above, the wife does say "I've got Shaun" when they're leaving the house. I'm back to believing that the spouse will have a fixed default name like the baby does.