„Jego” can refer to both, but for this reason it might be considered ambiguous. In fact, seeing a sentence like „On ma jego kubek.” I would sooner think that he has somebody else's cup. „Swój” as "his own" is more precise and quite common.
"jego" and "swój" are equally good answers because with the English sentence and a lack of context you can't really know whose cup it is: his own or some other guy's. So one of the options just shows as the translation a bit randomly.