The participle adjective "standing" or "standing up" means "Be in an upright position". This is how we would understand "He is standing"
The process of standing is very short. You wouldn't use the present continuous to mean "in the process of standing". I suppose you could use it to mean "repeatedly standing" (Compare this with "He is jumping") But that is a very odd thing to do.
If you really need to use a continuous form you could say "He is rising from his seat".
As the President was rising from his seat, a protestor threw a shoe at him.
Similarly "standing up" is a particle adjective and means "be in an upright state"
The boys were all standing up and chatting as the teacher came into the classroom.
This doesn't mean that the boys were rising from their chairs; it means that they were upright.
When you say "I am standing up" it does not mean "I am rising from my seat".