The player and his friends (Lilly, Nugget, Billy, etc) all have overactive imaginations and are just playing their own version of Cowboys and Indians, or in other words... They are just "roleplaying" or "playing pretend."
The teachers are overly strict, so they view them as the "bad guys," believe they are up to no good, and their imagination fills in the rest. Likewise, when real events (mundane or horrific) events happen in their lives, they adapt these events into their fantastic story as a coping mechanism.
The result are the missions that we play through in game.
Considering the first game was Monday and this was Tuesday, more than likely if another Kindergarten game was made, it would definitely be Wednesday.
Time doesn't progress until the Kid does an action or consumes an apple, and he's able to take items that he receives at the end of the day, back to the beginning of that day.
- Is Nugget also a being with time powers, too? It would make sense why he is behind the bookshelf after collecting all 50 monstermon cards, because when you piece together the ending that is canon from Kindergarten, the player hasn't collected the 5 nuggets of friendship in the Golden Ending. You also don't die whereas everyone else does, even Lily when you visit school. What the kid sees at school is Nugget's past self and behind the bookshelf is Nugget's present self.
We perceive kids as being hyper and overreacting. To give us, the player, this feeling, the game throws us stuff from way out of left field, like many horrible deaths. It would also explain why no one seems to be shocked, or even take notice of deaths (with the exception of Jerome mourning his father).
Another important note is how the days work. In the game, every single day starts the same, only to be influenced by your decisions and actions. Children don't have much autonomy over their environment. There are little to no long-term consequences for what they do, no matter how much of a big deal they think it is. That's why every single day is the same, and why none of your actions in the game ever carry over.
When you tell Nugget he killed everyone, he replies that "Nugget left Nugget and Nugget's friend alive". Note the singular. He's obviously referring to you, since you're clearly alive. This means Billy died offscreen, eliminating the Fridge Logic of why he was spared when Lily wasn't.
During the final mission, Bob is seen in there and you have to snap him out of confusion which then reminds him of the monsters in the principal's lab. He immediately grabs the axe and goes to fight off one of the monsters while you, Billy, Lily, and Penny all go to the principal's lab. The axe having blood on it indicates that Bob tried to fight off the monsters before school started, but the principal used the gas on Bob to make him forget about it. When you yourself are in the toxic gas, it only kills you likely due to you being a kindergartner and your body not yet developed enough to resist the gas, much like how babies can't be around adults who is smoking cigarettes.