The currents in Ampere that you measure this way are correct. After all you measure and display the currents in each main, not in a single consumer device.
The fact that these currents may run in and out of the same load is no contradiction to that measurement. You do NOT measure the current through one load but through each main of your supply.
In the end you usually want to derive the consumed power from the mains current. This is done by multiplying the current in each main by 120 V. (Ignoring Cos Phi for a moment - see below.) The power that you get by this multiplication is also still correct for your 240 V devices. For a single 240 V load you will get twice the current in the calculation, but then you multiply it with only 120 Volts instead of 240. So the end result remains correct.
About Cos Phi:
To calculate the power P from the current I and the voltage U you also need the cos phi (“power factor”).
This is the phase shift between the sine wave of the current and the sine wave of the voltage.
P = U * I * cos phi
For normal resistor loads cos phi is 1 and you can ingore it. This is true for many heating loads like ovens and radiators. But it is not true for inductive and capacitive loads like motors, transformers and most electronic devices.
With clamps alone you cannot measure the power factor. Instead you need more complex sensors that measure current AND voltage and the phase shift between them. Have a look at ATM90E32 Power Sensor for an example.