Davesprite is the only character actually stuck in the time loop. Cousin and all other characters are free.
At the end of Davesprite's route (assuming you never strayed), you see Cousin and Davesprite leaping into a white void together, implying that time is about to reset as you've made it back to the beginning of the game again. Attempting to romance Davesprite again will show that he remembers his previous relationship with Cousin. However, other characters' endings clearly show Cousin and their partner far in the future, such as taking a road trip with Hiromi and Toby (which they agreed they'd do after they graduate high school) and playing wrestleball professionally with Taira. Ergo, Davesprite is stuck (heh) in a single week of school, whereas time functions normally for everyone else.
Namco High is a school for bullied teens.
There are experimental schools specifically for bullied teens and children. These schools tend to have a higher than average population of LGBT+ students. Since every student in the game is potentially bisexual (and some, like Hiromi and Terezi, are explicitly not straight, regardless of Cousins' gender), this could indicate that Namco High is one such school.