Regarding the Fuuma Clan, there supposedly was actually a historical Ninja clan under the name Fuuma (or rather, Fūma) that existed. Far as I can tell it was eradicated around 1603 when their final leader, Fūma Kotarō was apprehended and executed. Don't quote me on this, though, as most of what I read is from Wikipedia as I only did a cursory google search.
Regardless, unless there is a TL note that states otherwise then it's likely that the Fūma Clan mentioned in the story is a reference to the 16th century ninja clan of the same name. Hopefully this eases some of the facepalms that a few readers may be having while reading.