...her father? her lost father that suddenly appears out of the blue at christmas to give her a present? dunno...

Being her Father would kinda make sense...

Hiding himself cos he was afraid to show who he was after being gone all that time.

Also, people in UK call Santa 'Father Christmas'

i relli don't think it's her father~

her father suddenly shows himself, Ayu's reaction would be shocked and sad rather than happy...

Hm... that is true, but maybe she was happy cos he was here to see her at Christmas, it is a time where families come together.