Wait what I'm confused. Isn't that the geography teacher? Did we get a timeskip in the mid...

Yes.
Yuri is in college now.

To be precise, in FYL, Kwon Yuri is a 2nd year student, with Moogyeong being in his 3rd and final year. Kwon Yuri and Yohan do have Geography Teacher as their homeroom teacher I think?

And he is a new staff, so at this point it seems like we have skipped forward around 2 to 3 years, where Yuri studies at the same college that Moogyeong studies at too. (Idk how long college is but it's around 2 to 3 years timeskip)

At this point probably teach has more experience teaching and is just getting jaded since no one listens to him and he probably still gets walked all over by students and staff becuase he is one of the junior staff and even his relationship doesn't go well...

Did the wedding happen while he's at university or was it when he was in high school? I just find the transition weird, he already knew the teacher way before but why is it as if they only just met now?

Probably becuase they were teacher and student for one year, and the wedding happened when Yuri was in university.

I think the transition should be seen as : Yuri and Teacher were in a homeroom teacher student strictly relationship, Yuri was being his usual "annoying" self with Geog teacher and Yohan (esp during the time after Yohan came back from the trip by himself) which was where something ties both Yuri and teach together.

Now after that, perhaps when Yuri and Yohan went to their third year, geog teach wasn't their homeroom teacher, and that's where the rs ended. Mayhaps Yuri changed a little after graduation, or we are just seeing more of his family dynamic as opposed to hearing it from Yohan only.

Unless you really do take note and have a more friendly rs with your teacher making it more of a rs where you treat each other as friends, students and teachers most of the time just pass each other by, hence maybe Yuri recognised geog teacher when he saw his identification card, but not during the wedding (becuase he was crying and maybe just more haggard looking and not feeling that great. The facial complexion does change. )

But weren't they quite close? Yuri even hangs out at the faculty room. I think seeing someone almost everyday for a whole year makes you familiar with their face in any angle. But yeah, I hope we get a better understanding with the next chapters. The way this one was presented was a bit weird.

People can forget faces pretty easily
Especially if it's been a few years
I forgot the face of my math teacher after not seeing her for about 3 years and didn't realize I had run into her when going back to my old school with my sister.
Same thing happened with my Bio teacher.
Plus, people age
He was sobbing so his face was most likely distorted a bit compared to Yuri's memory of him
Also, he wasn't expecting to see his teacher there so ofc it most likely wouldn't immediately register in his brain that this person who's sobbing is his old teacher

That's right. If you spend a few years later, people's faces can age and that will most likely make them look more tired than usual.
Yuri probably remembers teachers face as neutral/happier, so trying to recognise him when he was sobbing would be different. Plus I think that he remembered Teacher becuase of the identification card.