1. Andy has invited us to their party.
2. I called Chris and reminded them to bring their passport.

(Collins Cobuild English Usage 4th - 2019/12/1)

It seems that their means Andy in 1 and them and their signify Christ in 2.

Is this usage fully acceptable like It was me these days?


It means that Andy invited the people to someone else's party.

Mary is having a party. Andy invited his friends to Mary's party.

KeunChulLeeIt seems that their means Andy in 1

No, that's not it. More likely Andy invited us to a party that he (Andy) and someone else (his wife, a roommate, whatever) were having.

KeunChulLeeand them and their signify Christ Chris in 2.

No. It's at least two people, most likely Chris and another person.

KeunChulLeeIs this usage fully acceptable

The usage is not as you thought, but 'they', 'them', and 'their' can be used in a singular sense. This is mostly done only when the antecedent is indefinite (someone, no one, ...).

Someone left their sweater here. OK.
Robert left their sweater here. NO if Robert and their are intended to be coreferential.
Robert left his sweater here. OK.