Is there any difference in the usage of phrasal verbs ‘move over’ and ‘move up’ in those contexts?
There's room on the bench for six if you move over.
Could everyone move up a bit please to close the doors?

Student or Learner
Is there any difference in the usage of phrasal verbs ‘move over’ and ‘move up’ in those contexts?
There's room on the bench for six if you move over.
Could everyone move up a bit please to close the doors?
They can both be used to mean make space.
move up can mean shift to the next slot/position. You can use this when you perceive the space to be arranged into discrete slots.
There's room for six if everybody moves up.
move over doesn't have this extra meaning.
You're taking up all the space on the sofa. Could you move over so I can sit down.
I would associate "move over" to be a sideways movement and "move up" to be a vertical movement.
Could you move over please? I want to sit next to you.
Congratulations on your promotion! You are moving up the corporate ladder very quickly, aren't you?
In BrE, we also use budge up, when people move along a bench, etc, to make more room.
And "shift up/down" and "shuffle up/down".
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.