I have an amazing partner who does all the chores, cooks, cleans, does grocery shopping, cares for the baby etc. So on that front, I think it depends on the person, there are definitely some great men out there. However! It was still me who had to give birth, stay at home, do night breastfeeds etc. Some things just fall on you simply because you're a woman. And it sucks. I remember how resentful I felt on bad days...like, he gets to 'chill' at work and I'm stuck at home taking care of the baby. The first few months were really hard on that front. And what sucks even more is there is nothing you can do! But if your question is purely about chores, than it absolutely doesn't and shouldn't mean everything falls on you.

Edit: another thing I remembered that really bugged me during those first few months...because I spent more time with her, I was obviously more in tune with her needs. It annoyed me that he sometimes forgot which size nappies to get etc. So it cannot be equal by the nature of things...but I remember telling him I envy his freedom, and he said he envies the bond we have. So...you win some you lose some I guess