It's not exactly a birthday paradox because you have way more than 365 potential variants. Some 324 million. Plus its not discrete variants like having a single date of birth, you can have multiple combinations of variants.

But broadly the big utility is in matching distant relatives versus strangers from the lot, and seeing if there is agreement with those putative matches with actual records and other hard evidence. You share half your DNA with either parent, 1/8 with your first cousin, and so on and so fourth for the rest of the human race. Using that information coupled with other pieces of evidence can yield a pretty reliable hit and is how they managed to catch the Golden State killer using DNA and genealogy.

Does everything you just said not also apply to fingerprints? You leave them everywhere you touch, and people touch a lot of things. Maybe less so than DNA, since it's effectively a subset (things you touch likely have DNA from your hand skin, but things you don't touch might also have DNA from other things), but it's still just saying "you were likely here". And given a large enough population, wouldn't there also be many partial fingerprint matches?

I'm not sure how these two things are issues when comparing fingerprinting to DNA matching. They seem to be applicable to both.