50 ohms output termination for a 50ohms load?
If now we were to use the 50 ohms function generator for a 50 ohms
load and then measure the voltage drop on the load, would the
resulting voltage still be doubled?

No, the signal generator's 50 Ω output impedance and the 50 Ω load form a 2:1 potential divider and that means that the voltage seen across the load is precisely what the signal generator's display is saying it should be i.e. not double (as seen with a high impedance load such as an oscilloscope).
So, if the signal generator is set to deliver 1 volt RMS then, when connected to the appropriate load resistor (usually 50 Ω) then you will get 1 volt RMS across the load.