The light needs a live and a earth to light up, the live on one side of the bulb is supplied when you turn the ignition on, the earthy comes from the alternator IND connector, the thin blue wire, when the engine turns and the alternator starts charging this IND terminal also changes from negative to positive putting the charge light out as two positives don't make a bulb work(no circuit).
It is entirely possible and has happened millions of times the alternator charges perfectly but the IND remains negative, just unplug blue wire and see what happens, if so a duff rectifier/regulator part on alternator, which you can either live with or not.
tabetha