Yeah, damn, I tried to play Day 2 (I'll tell you in advance that I saved my choices correctly in the slots during the playthrough) and then they tell me to enter my gender and name again... but in general, nothing changed during Day 2, as if everything was the same as Day 1, what's the problem??
So... As soon as you finish Day 1, you go straight to Day 2. The button in the menu for Day 2 is for when you don't want to play Day 1 and want to skip straight there. IT'S NOT A BUG.
In fact, Day 1 ends when you are kidnapped by Pierrot or leave the circus for the first time. If you've played beyond that point, you've already experienced Day 2. Yes, the game is a bit short, only 2 hours of gameplay for both days combined.