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.