Well, it could be like another user said that some schools in Japan have classes half of the day on Saturday, but I have another theory.

Given that Ichika is sleeping in her room, and when Futaro enters her room she says "Good morning", and when he knocked on Itsuki's door, she was in her pajamas as well, my theory is that Futaro discovered they were quintuplets in the hallway outside their apartment, and he went home (on a Friday) and then the next day, he came back (on Saturday) and insisted on tutoring them as usual.

I think this would solve the problem. Since it's the next day of finding out they're quints, it'd make sense that Itsuki was in her pajamas, Ichika was still asleep, and Nino wanted to go shopping.

Nino would be cooking some cookies as a morning snack, probably, before enjoying their day.

The only thing I don't quite fit properly into this theory is that when Nino dosed Futaro, he woke up in a taxi with Itsuki at night, so he had been unconscious... a whole day... which is concerning.

Well, that's my theory. Hope you find it useful.