Does a unit being transported with a standby token keep the token when the vehicle transporting it move or is the token removed as the unit is considered moved when the transport moves?

I would say that it keeps it as it has not done an action or received any suppression but I'm not 100% sure

it would keep the standby token.

you can spend a standby token to disembark as well

The unit didn't do any of the things that would cause it to lose a standby token, so it keeps it.