Searching a big DB is one way to use fingerprints. It is a crappy way, though.
Another way fingerprints are used is to compare fingerprints found at a crime scene to people who have been identified using other methods that have nothing to do with fingerprints.
E.g., if you have someone murdered in his office when working late at night, and security cameras show that five other people were in the building at the time of the murder, and you are able to get a good set of fingerprints off the murder weapon you don't need a database. You take the fingerprints from all five of those other people, and if one is a very good match and four do not match, you concentrate most of your effort on the one that matched.
It's actually quite similar to how DNA evidence has been used and misused. I don't know how they compare DNA nowadays, but when DNA evidence was first making waves getting people convicted they only compared two DNA samples at a few points. For a given sample there would likely be several people in the country that matched.
That's fine if used right, like fingerprints in the earlier murder hypothetical. Narrow it down to only 5 people who could have committed the crime, get a DNA sample from the crime scene that must be from the criminal, and if that matches exactly one of those 5 suspects it is strong evidence they are the criminal.
Have no suspects yet, run that same sample through a database, get exactly one match, and conclude that must be the criminal. Totally bogus. A lot of people were convicted in the early days of DNA matching that way.
The database method can be made sound, but only if the database includes everybody. Match against a database that includes everyone, only get one hit, and you've probably got your criminal. But if the database includes everyone you are likely to get several matches.
As I said, that was how it used to be. I know DNA sequencing has gotten faster and cheaper over the years, but I have no idea if routine forensic DNA matching now matches enough to make matches unique except in the case of identical siblings.