The simple answer is that the word is used in a specialized context here, wherein it is almost always used in the plural. (Here's a little evidence from NGram Viewer). So why is it plural? It just (usually) is--you'll see it both ways, but plural is more common.
That said, some schools insist on using "Admission" instead of "Admissions" (including one of my former employers, who had me do a lot of picky edits to some form letters...). It is less common to refer to "a university department that governs which students are admitted" with the singular word "Admission", but it is done sometimes.
As for why it is plural (beyond "it just is"): I am not finding anything in my usual etymological sources, so the following is speculation on my part: the process of seeking admission is named after the administrative department that decides who is admitted. This department is in charge of "admissions". Whether you interpret "admissions" as "students chosen to enter the school" or "permission to enter the school", the department will be dealing with admissions in the plural--it will choose more than one student to admit; it will grant more than one right to enter. You can think of it like how in a store we have a "Department of Returns and Exchanges" rather than a "Department of Return and Exchange". The store's customer service does many returns and many exchanges every day; it isn't dedicated to the idea of "return" and "exchange" in general (the way that, say, a stock exchange is).
But again, some schools call it an "Admission Department" or an "Admission Process" because they're thinking of "getting in" in the general sense, as opposed to multiple instances of one person's admission.