When this happened to me I just provided a fake (photoshopped) ID with my pseudonym and they accepted it.
To this day she has an account with her name misspelled.
If you are issued a document and then manipulate that document (especially if you do so with fraudulent intent and then pass it on as authentic), you are committing Urkundenfälschung ("forgery of documents") which can be punished with up to five years of prison or a monetary fine.
This applies to everything from government-issued identification to sick notes. The exact punishment likely depends on the intent and the type of document in question but forging government IDs is almost certainly a crime.
OTOH Facebook would have to obey certain laws if they want to check your ID, especially as they ask you to send you a full unredacted copy. Germany has a law book called the "Personalausweisgesetz" which defines who may ask for ID and how it has to be handled. That's aside from the privacy laws which already define rules for companies that handle or store personally identifiable information.
It's not entirely unlike having a friend show up in a fake police uniform to vouch for you.
There are edge cases like art or educational purposes, but using a fake ID to prove your identity to a third party is definitely fraudulent: you're saying "this authority attests that this is my identity" when the authority in question does nothing of the sort and the document claiming it does is forged.
If you're a threatened person, and yet you still need to be on Facebook...then the time it takes to Photoshop something will be insignificant to the apparent benefits you get from Facebook. For the troll? Not so much...getting to be an asshole is not the strongest incentive to jump through the hoops.