"Employee" in the phrase "employee ID" is a determiner, not an adjective—right?

It's a noun modifier. See the Wikipedia article on grammatical modifiers for details. Whoever wrote the example saying that "employee" is an adjective in the noun phrase "employee ID" has been confused into thinking that any word that modifies a noun is an adjective.

It's not a determiner, though, either.

Determiners form a closed class in English, so I wouldn't say that employee is a determiner. "the employee id" would be impossible if it were a determiner, too.

On the other hand, I wouldn't call it an adjective either. My reasoning comes from examples: a pretty person is pretty, but employee identification isn't employee. No, employee is a noun, and, probably through an adjunct, modifies the other.

How could it be an adjective? That would make "dog" in "dog food" an adjective.

Funny… Almost two years later and I am considering a similar thing (what employee is to employee first name) because I am reviewing a naming standard. First is an adjective for name and most times that is all you may need. However you MAY need more in large enterprise data models. Wikipedia seems to suggest that employee in this data name is a noun adjunct—a noun that modifies another noun which they say "functions as an adjective". I like this final wording; it's easy to understand and I think that it is what I will use in naming standards.