Professional technique used for logging power consumption of a DC circuit
Current measurement is mostly converted to a voltage measurement. The reason behind this is that very precise and stable voltages are possible to build based on semiconductor physics (bandgap), but current references with such precision are not available.

There is no need to log two channels. With a difference amplifier a floating voltage can be measured and converted to single ended signal.

DC power supplies have built-in current measurement capability. They accomplish this usually by having a resistor in a feedback loop, so that the voltage drop on this resistor does not alter the output voltage. This resistor can be  programmable to alter the resolution (uAs vs As).