Once, or once per activation
Adopting the “there are no hidden rules” rule, since there is no suggestion that multiple saving throws are required, they aren’t.
A creature “that sees its reflection in the activated mirror” makes “a DC 15 Charisma saving throw”. The previous paragraph spells out how the mirror is activated.
A person sees their reflection and from that point on they have seen it, even if they are still looking or are looking again or looked 30 years ago. Once you “see” Paris, you have forever after seen Paris even if you live there.
Whether the mirror “resets” when deactivated/reactivated is ambiguous so that’s a DM call.