You are absolutely right in saying "not everything that describes something is an adjective".
The finger part is not even "functioning like an adjective", which seems to be a rather popular weasel wording (and not just on this page). It cannot be used attributively, it has no comparative, and I'm forgetting at least one other thing right now. A fast car is fast, and a beautiful woman is beautiful, but a finger print is not finger. You can have a faster car and see a more beautiful woman, but you won't find a more fingerprint or a fingerer print. It is not an adjective, is not functioning like an adjective, and has nothing to do with adjectives at all.
The current lingo would be noun adjunct or attributive noun. I say "current" because part-of-speech definitions are not set in stone and keep changing every couple decades. If you ask five people you will get six different wordings, and frankly, nobody prevents your friend from just making one up on the spot. They actually can make the definition be "everything that describes something is an adjective", and have it include verbs for all we care, because verbs describe things, too. In "I ate a giant sandwich", the ate describes what I did to the sandwich, and in swimsuit the swim describes the suit.
The elephant in the room is that it's really irrelevant what they call it. It's just a label that doesn't miraculously change anything about how the language actually works. The finger in fingerprint will still be different from the black in blackboard, the swim in swimsuit, the parking in parking lot, the well in well-being, or the off and on in offset and onset. So if they want to account for that, they'll have to invent subclasses for their "adjectives", putting them right back to square one.