IIRC FN sold the same gun under two markings, as a Browning in the US Market (Baby Browning) and as just FN in the rest of the world.
I believe you are correct. This is my FN 1906 vest pocket.FN does not label their stuff Fabrique Nationale. It should say Fabrique Nationale d'armes de Guerre.
Like Colt doesn't just stamp Colt. They mark theirs Colt patent firearms manufacturing (Colt Pt FA Mfg).
There is at least one company in Spain that calls itself Fabrique Nationale.
Just a gut feeling, but I think you have a fake.
Disassemble the gun and take pictures of the proof marks. I would be very surprised to find a Belgian made gun that does not have at least a half-a-dozen proof marks on it.