>> one of the next events (prince route) is that he gets to know her parents before they even have had a date
This is actually a random event, not a Prince story event. It can occur at any point beyond the common route, and they can happen in any order.
>> why can MC check in what there are doing?
MAJOR SPOILER: The MC works for the government security department. The game's world is not the same as our own; it's more dystopian.
>> she casually just kisses him
Past the common route, they are, in fact, dating and technically in a relationship. The MC just hasn't 100% committed to Boyfriend yet. Keep in mind that each in-game day that passes, although there aren't scenes for all 100+ days, they are actively furthering their relationship (with some exceptions).
>> that has to be a bug, hasn't it?
The word in question is actually "redacted". It's not a bug; it's a reference/hint to the fact that MAJOR SPOILER: MC's job requires them to be secretive about their personal life.
>> suddenly was sure that she is in love and
I double-checked, and the MC actually doesn't say they love Boyfriend in the particular scene, although there are references to them beginning to fall in love with Boyfriend for some personalities, which is accurate.
No problem!