Yes, university and school have similar uses - in most circumstances.
The reason why the use of school and university may differ is that the time spent there is structured differently. This means that the idiomatic use of after school and after university are different (try searching for them).
After school - the time of the day after school has finished (see after school clubs, for example)
After university - the time after you have graduated from university
When attending school, students are in the school site between set hours that are always the same, and they generally can't leave during those hours. The teaching is all very structured and the school site is generally relatively small, and not where the students live (unless it's a boarding school). Therefore, during the day, there is a very definite at school and not at school.
University tends to be somewhat different. Often students will live on the university site, or in university owned accommodation. The teaching is much less structured (more of the time is self-study), is often less regular (one day lectures could be 9-11, the next may have no lectures, but a seminar from 2-4) and is generally more diverse (there are lectures, seminars, labs, etc). The site is much bigger and either will be dispersed throughout a city, or will make up a campus that a student might not leave very often (because they live there).
It therefore often doesn't make that much sense to say "after university" in the same way as "after school", because that could mean many different things and refer to many different times. Many students will not consider themselves as leaving university at any point during the day. You are much more likely to hear "after lectures" or possibly "after class" (although I think of that as being more of an American phrase). You might also hear something like "See you after I've finished with uni (work) for the day".
That said, there is nothing to stop you from using university to mean the time during the day during which someone is at university, you would just have to make it clear from the context that that's what you mean. It would generally require more explanation than saying "after school" would, and there is probably a better way of saying it.