My understanding is that modern aircraft measure the fuel volume and temperature and then calculate the weight. Is this right?
Short answer is yes - the aircraft measures the volume of the fuel and the fuel density and uses that to calculate the mass of fuel.
BTW, while the energy per unit mass of jet fuel is close, it's not a constant - there is some variation around the mean, although it's pretty much in the mud compared to other variables like engine-to-engine efficiency and airframe drag differences.
Back in the old hydro-mechanical fuel control days, there were adjustments on the fuel controls (adjustable on-wing) for fuel density. My understanding is the adjustment was seldom used, but it was available.
Today, with FADEC, that density adjustment has gone away. The FADEC controls closed loop to rotor speed or EPR, and accels/decels are to rate of change of rotor speed (normally N2dot (or N3dot for Rolls), although N1dot does come into play sometimes). There are maximum fuel flow rates based on fuel metering valve position during transients (weight fuel flow/burner pressure - often called "Woof/PB") but those rarely if ever come into play.
About the only time fuel density might have a noticeable affect on FADEC is during initial lightoff during starting. FADEC uses a fixed fuel metering valve position - which will result in a fixed volume of fuel - until the engine lights and starts to accel. That's one of the reasons it can be hard to start when it's really, really cold.