I think it can happen very, very easily.
With the traditional set up, of mum taking maternity leave, she will taking the primary role of caring for the child, when very young, because she simply is the one at home doing so.
If she's breastfeeding, then she'll be doing either all of, or virtually all of (if also expressing) the feeding, and a lot of the comforting of the child.
Yes, dad can come home in the evening and do stuff, but, if he's not doing the days, and not doing the nights (especially if he can't because of feeding), then he's not going to be as in tune with the child, and their routines. So the child is more likely to start to go to mum/settle better for mum, and the dad is more likely to ask for guidance on how things should be done etc, because they do it less often.
Then women are more likely to go part time, which again means that the woman ends up more as the default carer, and the child will be more used to her sorting him/her out, than the dad, because mum is around more.
And then one day, you realise that you've totally fallen into a default parent situation. I think often its not delibrate on anyone's part, but its how the dynamic can often end up being, unless people make real efforts to redress the balance.
As much as you take on that role, you assume that role yourself by telling him when you are going out etc. The fact that he doesn't, probably isn't him thinking that he's not in charge, but if the kids don't tend to bother him to ask him things, or come to him as much, then he may simply not think of it.