You're correct, hence the "essentially" qualifier. Technically beat 'em ups existed before Final Fight (River City Rampage existed before either of those games), but it wasn't until Capcom's game that the genre became codified and started churning out endless amounts of derivative cash-ins.
But I wanted to point out that Double Dragon definitely pre-dates Final Fight.
I always loved that Hagar in Final Fight was Metro City's MAYOR. I bet he saves the city a lot of money on police overtime.