In many cities you can share your metrocard (multiple persons validate the same card when getting on the metro or bus). Is this allowed for the metromoney card card?
No, you're not allowed to do this, given that the card always requires a minute between validations, and there are guards at the barriers to see to it that there is no abuse.
However, if you save the original receipt, you can get the 2 lari back when leaving Tbilisi (if less than 1 month after buying the card), by presenting your card, receipt and passport at a ticket counter
Seeing that the card itself costs a mere 2 lari - which is less than $1 - I don't see why you would need to share one. Simply get two (or more) cards for each person and top them up individually. A single ride is around $0.20 so it won't break your budget.