Does a flex shaft need to be on a stand?
No, a stand is not required but it does give you some more options. A stand that you can clamp the flex shaft tool into can give you hands free access to it, like a mini buffer/grinder. Even without that possibility, it gives the tool a secure place to rest when between uses. 

Keep in mind that it is a relatively heavy tool with a spinning chuck on the end of it and connected a long thick and somewhat springy wire/tube anchored somewhere else. Without the stand it could (will) flop around from momentum and is going to want to find its own resting position that may not be the one you want. With the stand you know where it is when you need it and its a lot less likely to go off in its own direction.