The Tama wears the costume until it goes to sleep. I think what's happened is that your Tama's sleep has coincided with a normal 'Adult leaving baby' time as well.
It's just a coincidence that your Tama was in the costume.
Normally what happens is that once the adult flies up off the screen at midnight, the baby is left sleeping - and the parents details ARE still there.
They stay there until the baby wakes up. Once the baby wakes up on it's own, it starts crying and the first button you press (any one I think) starts the sequence to give the baby a name and then off you go!
Best, etc.
TamaMum
PS Goodness - you were up early this morning....