Is there a single-word category for the status Freshmen, Sophomores, Juniors, Seniors? I've heard it called as "batch status" but can't seem to find the right single-word category for it.
You might consider class as a generic noun that covers these terms.
All members of the junior class will become seniors next year.
David is an upperclassman now that he's in his third year at Brandeis.
But despite what others have stated, I believe the term "cohort" has a slightly different connotation. It means
Cohort (educational group) A cohort is a group of students who work through a curriculum together to achieve the same academic degree together. A cohort forms when the students begin the curriculum and typically does not admit new members afterward. [Wikipedia]
Here's another reference for cohort:
Cohort study is when a group of students, 12-25 (this is a general number that may vary by college), start and finish their degree together. They are accepted into the same program, take their classes together, and graduate together, building relationships with one another as they do.
This would seem to invalidate the notion that cohort could be used. Typically juniors, seniors and the rest do admit new members (transfers, etc.) and they don't all pursue the same academic degree.
We don't really use any of these terms in the UK, so maybe I'm not qualified to answer. But I think in general the only significant subcategories within tertiary education are first-year and final year students.
Most courses tend to be 2-4 years, within which it often makes a difference if someone's just starting or about to finish, but there's little reason for most of us to distinguish any particular year-group somewhere between those limits.
One group who are likely to need to refer to specific year-groups are the staff at the educational establishment. In the UK they probably normally do this by referring to the 2009 intake, for example.
I'm a prof at a US university. We (I and people I hear from on this topic) call these classifications (freshman, etc.) "rank."
We don't use "cohort," because that denotes a group that takes limited-availability classes together.
We don't use "class," because that denotes the year of graduation, not the current progress toward that year: "class of 2020."
We don't use "grade"; that works in the US up through 12th grade, in high school, but colleges don't use it.
We may say "year": "What year are you in?" maybe answered "I'm a junior." I think this is informal, and "rank" is the usual term I hear, but "year" can work too.