The context shapes the interpretation.
For instance, if your example sentence was amended by dropping the word 'enrollment' from the tautologous term 'registration enrollment':
During last year's registration, we had 1,457 children register
it is clear that here, 'registration' must refer to the process or period during which the entire population of pupils was being registered.
It would therefore be incorrect to say that in this context, 'registration' refers to the registration of an individual student.
However, if we take a different sentence:
I need to finish typing up Amanda Smith's registration before I go to lunch
it is obvious that we are talking about the registration of a single individual.
Incidentally, application, admission, registration and enrol(l)ment are all possible options here, depending on which jargon term is used or preferred by the relevant educational institution, and in some cases on whether there is a screening-out process. (For instance, application tends to imply that not all comers will necessarily be accepted.)