

mAh/g (milliamp-hours per gram) is essentially still a measurement of capacity, but in terms of current instead of power.
We can do a little dimensional analysis here to translate between them. Power = Current x Voltage, so you’d multiply this (Current x Time)/(Weight) value by the nominal voltage of the cell to get to (Power x Time)/(Weight).
Phone batteries are often specified in units of Current*Time (e.g. milliamp-hours), but I’m not completely sure why. I think it has to do with voltages being standardized for certain types of cells, so the only real variable in the battery capacity is the current.
Edit: rearranged some ideas to make more sense
Realistically the connector would have been proprietary, but I can see a world where we got cartridges that came in little cases like the games for nintendo ds or switch.