My best guess is that the government knows, but they might not care, especially in this specific situation.
From what we've seen, the government in the Incredibles universe actually cares about the Supers. They work hard to relocate them and keep their identities secret, and from some of the interactions we've seen (especially the scene with Rick Dicker in the limo towards the end), it seems like the supers are actually old friends with the National Supers Agency/Super Relocation Program members. With this in mind, it would seem that the government wouldn't want to needlessly get in the way of Supers. So if a Super is actively committing crimes or endangering lives, then of course the government could probably figure out who and can step in.
In this specific situation, the government probably knows more than the Police that first arrived at the scene. The police just saw two masked men in a jewelry store, apparently using the fire next door as a distraction. After Frozone freezes the police officer, I would imagine that the Super Relocation Program would somehow hear about the unauthorized use of superpowers. Upon investigation, they would identify Frozone based on his powers, and it wouldn't be too hard to guess that the very large masked man with him was Mr. Incredible.
However, with the number of civilians that they saved from the building, and the fact that neither of them (probably) has any history of criminal behavior, the government would be able to come to the conclusion that the police couldn't make in the heat of the moment: The two masked men were there to save lives, and simply ended up in the wrong place at the wrong time. Thus, they wouldn't pursue any action beyond a reprimand against him.
We also know that, while the officer is frozen, he is for the most part unharmed, as we can see his eyes moving back and forth even after being encased in ice. He can simply be thawed out and be as good as new; the Super Relocation Program even has the resources to wipe his memory of the whole affair (as seen in Jack Jack Attack), to avoid any future potential actions from the officer against Frozone.
TL;DR: While they probably knew that one of the masked men was Frozone, they also knew enough to realize that he wasn't trying to be a criminal.