Each GL's powers rings are assigned to the chosen GLs, but power batteries are common, meaning that GL's can have more than one power battery, and they can give their Battery to other GL's.
Hal was able to charge his ring using Ship's power battery, so any power battery can be used to charge up the ring.
I think that Hal had kept his power battery at Ferris air base on Earth, and taken another power battery with him on the ship which was an experimental power battery (as Ganthet had told Sayd in the opening before he was exiled from Oa, that Battery was capable of turning into a Blue Lantern Power battery, which finds Saint Walker.)
(Note: In GL comics, power batteries are kept in a unique pocket dimension, so that GLs can retrieve them no matter where they are in the Universe.)