You generally can't actually check the government databases directly, but you can still determine this.
First, companies can catch most fraudulent documents simply by looking at the document (eg. are the fonts all correct, does the checksum on the MRZ add up, does the data in the MRZ match the data on the face of the document, does the data on the document match previously collected data about the individual, etc.) Some will go further by combining this with a "liveness" check (eg. they might ask you to take a picture of yourself in a certain pose, or to record a short video looking side to side)
Second, companies can use a soft credit check (if authorised by the user, which would need to be in the fine print when you sign up or when you are asked for such a document). Such a credit check won't affect your credit score, but can be used by companies to see if an individual with your details exists. Companies which offer such credit data in the UK/US/other western countries typically boast of 90-95% match rates across a population, but obviously younger people are less likely to be found since they are less likely to have a credit history. This is typically aggregated with data from non credit sources (electoral roll information, county court judgements, etc.) to reach those high match rates. They might also geo-locate the IP address from which you accessed their site and compare it against any address information they have on you (which could come from you providing it on sign up, it could be extracted from the document if it's a driving license or something, or it could come from any credit records they found relating to you)
For Facebook specifically, they might look at other online activity - other social media accounts they can link to you, etc. And throw all of that into the mix.
If at the end of all that they don't have a clear answer, they might fall back to a manual process, or allow the account to be created but have content posted by the account flagged for manual review.