It’s not referring to fish, though the placement of the anatomy fits. Fin was defined in the 1796 British Vulgar Dictionary to mean an arm.
It’s actually still in slang dictionaries and thesauruses today. Along with flapper, flipper, rudder, and wing.
There’s more slang for the muscles of the arm, the biceps in particular - guns (as in the gunshow while flexing) and pythons.
But sometimes the arms are simply appendages or upper limbs.
The Cant Dictionary also defined the arm as a rammer or smiter.
Do you have good slang for the arm?