But does that also apply to prints collected for government ID cards?

Most probably, for several reasons:

1. If the government or a goverment organization wants to fake the fact you've presented your fingerprints somewhere they can just fake the results of the checkup itself. And if they're up to this level of fuckery it's probably a short distance to where they just imprison or kill you, so having your prints faked is the least of your problems.
2. If the goverment is well-meaning they don't want to store fingerprints because they're not needed and they'd just be storing highly sensitive personal information that, if ever breached, could be used for all kinds of shenanigans. The best way to protect data is to not have it in the first place.
3. The goal of these systems is to log and attest the checks, not the fingerprints. They document the fact that at a certain time and location the checksums for a set of biometrics did or did not match some reference checksums. They don't care what those biometrics mean, or what the result of the check being passed or failed means, or what the actual biometrics are (we're talking about fingerprints here but there's lots of biometrics that can be used).
4. Storing actual biometrics would take a lot more space and add complexity. The checksums are much smaller and simpler.