Usually top level domain is abbreviated TLD, but that's not important. You are correct that a TLD can be a country code or one of select list.
A lot (most?) academic institutions tend to create a sub-domain under their ctry code, because it makes sense ie indicates to people where they are based and the likely lang of the website in qn.
Most do not have external students in other countries.
Depending on Registrar's rules/checks(!) you can have almost any TLD name you want ...
For sub-domains, the TLD owner would have to agree as well.