Laptop shows battery is 4% charged, but it is charged to maximum
You seem to be confusing battery voltage with actual battery capacity.

The charging system may be doing coulomb counting to determine state of charge. As such, it would actually measure the amount of power going into the battery. Considering you say the laptop is old, it's possible the battery can only store 4% of the power it could when new.

Basically, measuring the voltage only tells you if it's currently fully charged, not how much energy "fully charged" represents. Coulomb counting can track actual capacity.