I went to university in Ireland. In the alumni magazine people are identified by year of graduation, but nowhere else. While I was there it would have been regarded as bizarre - people were identified (if relevant, which it mostly wasn’t) by the year they were currently in (first year, second year . . . or more formally Junior Freshman, Senior Freshman, Junior Sophister, Senior Sophister). Graduate students were never identified by year; you were just doing a master’s in ignorance-fighting or a PhD in straightdopery, or whatever.
That’s what my HS would do. Didn’t always work.
But it’s high school – fits right in. 
I can confirm that for Germany. In high school, holding back, which compels the student to do the same year again (i.e., now become class mate of a cohort that used to be a year below) is not uncommon. As for tertiary education, until relatively recently university programs would not even have a fixed duration; your degree would take you whatever time it would take you, and some students were faster and some slower (there would be a recommended duration, which might matter for things such as government stipends, but it was in no way mandatory to complete the degree in that time). More recently, after the transition from the old German “Diplom”/state examination system to the bachelor’s/master’s system (UK style, with three-year bachelor’s) this has changed, and degrees now have a much more rigid schedule than the old “I’ll do this semester whatever classes I feel like taking” approach. But this happened after I graduated, so I don’t have first-hand experience.
Yeah. The MIT class notes have undergrads sorted by class - graduation year - and graduate students sorted by department.
In Whitby, Ontario, in the ’70s, ‘public’ school went from kindergarten to grade 6. ‘Senior public’ school was grade 7 and 8 (typically, ages 12 and 13). Both of these were formally called ‘primary school’, and you got a certificate/diploma with a graduation ceremony at the end of grade eight.
High school (‘secondary school’) was grades 9 to 12, with an optional fifth year of high school, grade 13, for students going on to university.
You got an Ontario Secondary School Graduation Diploma after grade 12, and students who took grade 13 got a second diploma, the Ontario Secondary School Honours Graduation Diploma.
In both cases we referred to the final year we graduated; a student who started grade 9 in 1976 was class of 1980 when graduating grade 12, and class of 1981 if graduating grade 13.
University courses were set up to assume incoming students had had grade 13; I believe that students from other jurisdictions without the equivalent of grade 13 had to take an additional year of university courses, though I am not certain.
Later, grade 13 was eliminated, and its contents were spread among the other grades. I think. I do not know how this affected university programs.
We didn’t, and still don’t, use the terms sophomore, junior, or senior, in high school or university. There is some limited use of freshman (‘frosh’) for both grade nine and for first-year university students, but even then it was more usual to say ‘grade nine’ or ‘first year’.
In Ontario, college was (and mostly still is*) separate from university. University has four-year degrees, graduate school, and original research; college has one-year certificates and two- and three-year diplomas and is more geared to training for specific jobs. I have a three-year Electronics Engineering Technology diploma from a college, rather than a four-year electronics engineering degree from a university. I gather that there was a lot more practical work in the technologist program than the engineering degree.
* Some colleges have a limited number of four-year degree programs and even research, often as part of an attempt to become a university.
I enetered college in the fall of '93, which should have made me class of '97. I took the better part of a year off mid-way through my junior year, so I ended up graduating in '98. I consider myself class of '98, but the university, for alumni relations purposes, allowed me to pick either '97 or '98 as my class. I just stayed with 1998, because that is the year I actually graduated, and it seemed silly to me to list myself as class of '97 (though I suspect any “official” printed announcement the alumni magazine prints would have put me as '98, but I don’t submit any career notes or life milestones or anything like that for publication. Actually, now that I think about it, I don’t know how much they fact check it, so if I identified as '97, perhaps they would have written me down as such.) The good news was that, being class of '98, I got to see and hear Ruth Bader Ginsburg give the commencement speech. The class of '97 got Bill Cosby.