Can I call that person a schoolmate? Or is that a wrong word because a school isn't a university? In that case, is it common to call that person a university mate?
In the US at least, this person would be a fellow student, a friend from college or a friend (or acquaintance) from school.
Personally I've never heard "university mate." Also I've never seen or heard "an university" anything. It would be "a university" something. Because "university" starts with a Y sound, a consonant in this case.
A "classmate" is only for a student who's sitting in the same class with you.
I haven't heard "schoolmate" but I don't know why.
A person who attends the same college or university as you, from a more technical perspective, should probably be called your collegemate (college is more or less a general term for an institution of higher education, at least, in North America) rather than your schoolmate, but I wouldn't say that this term is common enough that you will ever hear someone actually say it in real life. I've definitely never heard anyone use it. Though, technically speaking, it does exist in English. More realistically, you would probably just say something like people who go to the same university as you or people who study at the same university as you or simply fellow students from my university.
The word classmate, on the other hand, would be a term that's used to refer to someone from your class regardless of the kind of educational institution that you're attending. It can be a high school, college or university. Your classmates are simply people who are in the same class as you.
A few comments from the UK. "An university" used to be current 200 years ago. See for example Google ngrams. Also I recall seeing more than once (but cannot now trace a reference) the definite article being used at about that time, as if there were only one university. (There had been two in England for centuries). "Uni" is now very commonly used in spoken BrE but is newish. The older informal term abbreviated the word from the other end: Varsity. So, for an older generation: the answer would have been "a Varsity friend".
In formal BrE, "at college" is potentially confusing because there are lots of colleges in the UK that are not universities - and there used to be many more that have now become universities, and some of them were previously known as "university colleges".
The AmE use of school to mean university is also confusing for us.
And all the more confusing for everyone is that both 'college' and 'school' are used in BrE to refer to aspects of universities. When I was an undergraduate I was a member of a college which was not in itself the university, and I am now associated with the School of Mathematics and Statistics in a British university, but that school is likewise not a university in itself.
What this all adds up to is that there seems not to be a single word answer in BrE. I personally would use "fellow student".
Since nearly every university in the U.S. has a school nickname, it's very common and well understood to call other students by the nickname. This reference would cover students attending as well as alumni.
For example, if you and your friend go to the University of Georgia, you could say, "Michelle is a Bulldog too."
We went to the pre-party and the bar was full of Bulldogs.
I've never heard the term schoolmate. I've heard classmate, but that is specifically for someone in the same course as you. Where I grew up (near San Diego), we would generally refer to other students at our same school as peers. Since peer is a very general term, you could also qualify it, like school peers or class peers.
In Germany we have a special word for that "Kommilitone", opposed to "Mitschüler" what means schoolmate, or "Klassenkammerad" what would be "classmate"
so let's see what auto-translate-tools get for that...
so I'd say "fellow student" seems the most correct choice, even though it's not "a single word"
Yes, schoolmate is perfectly acceptable. At least in AmE, where school is used interchangeably with college or university in informal speech. In BE, it seems like it would not normally be used this way as school is normally not used beyond secondary education.
1. a companion or associate at school.
As noted in some other answers (which specifically disagree with using this term in this way), using the term "classmate" can be a bit controversial. If you are in at least one class as the person, then the term is applicable. In saying this, I am using the word "class" as meaning a group of people who meet at the same time, under one instructor.
However, even if I have someone who started at the same time as me, and graduated at the same time as me, we could still be classmates. For instance, a person born around 1977 may have graduated high school around 1996, and was part of the "class of '96". (Likewise, then, those who got a bachelor's degree four years later would be the "class of 2000".) If I wanted to refer to such a person, I would feel right in saying that we were in the same "class", since the word "class" has multiple meanings.
Note that if the person is one grade ahead or behind you, you may still attend college at the same time, while not being in the same class. So this term would only apply to some of your co-students.