If the responses are actually anonymous (as they claim, but I don't believe it's actually anonymous in their database), it would be easy to generate enough false positives to spoil the system.
I personally had my name in Russian letters, so that it resembled my real name with symbols. Facebook got aware of this and blocked my account and forced me to upload a scan of my ID card. After a heavy dispute because I was not willing to hand out my ID to a foreign company (and various other reasons) I gave into their request, uploading a fake ID of me with fake data.
There is a 0% chance that Facebook does not record your response to every request.
If it turns out it isn't anonymous and they are shutting down accounts over it, can there be any repercussions for them lying like that or is it just "unethical"?
Funny - it says they won't take any action over it. Do I trust that?