From Wikipedia: "Unlike many other jurisdictions, the New York City fire marshals are armed police officers with full powers of arrest..."

https://en.wikipedia.org/wiki/Fire_marshal#New_York_City_Fir...

Almost every fire marshall is a licensed peace officer (e.g. arrest power).

I live near a city which only has a single police/fire department, and all officers are both policemen as well as firemen (including the police chief).

It's probably a fire marshall/inspector, they are usually under law enforcement, but not employed by the police. They are hired either directly by the municipality; fire department; or County/State. Since they handle fire laws; fire safety; investigations; and fines, it makes sense.

Edit: Clarity.

Actually everyone is capable of arresting anyone else. It's called a citizens arrest. It is legal but risky (both physically risky and legally risky).

I have once performed citizen's arrest — I would not recommend doing this unless you want to visit the courthouse multiple times (as a witness, hopefully).

I don't know about the FDNY in particular, but lots of agencies beyond FBI / police have jurisdiction to arrest people.