When Alaric first met Hope in Voodoo in My Blood he said they opened school with Caroline and now in the latest episode of Legacies where Lizzie and Josie are in Alaric's mind, we see that Hope was recruiting into the school even before it opened. This should be considered an continuity error.

Also, from my perspective on the Originals, Alaric wasn't so close to Hope, he was just her headmaster and some promotional materials also said that Alaric became a father figure to Hope after her parents died. When Hope was dying of Hollow, Caroline cared much more for Hope than Alaric, as if suggesting that they had some closer relationship. When Rebekah sent her to Klaus and they talked about Hope, she seemed to have a pretty good insight into Hope's life. I know everyone will say that we haven't seen much school on the Originals, let alone Hope's relationship with her headmaster and headmistress. However, the truth is we saw more of the school vibe than in Legacies.

Honestly, I like Legacies, but in the last season of the Originals, the Salvatore School looked more like a school but in Legacies it looks more like Alaric living in a big house and has adopted a lot of kids. The best example is from the last episode, Lizzie asleep on the couch in the lobby opens the door to the police, not some random school worker or teacher. Besides, it seems strange to me that someone is knocking to school.

All in all, a school that lacks the boarding school vibe and retconed Hope's entire history at school from the separation of the Mikaelsons to Legacies Season 1 (including Hope's relationship with Caroline, with other students and when she began attending there).