I am currently storing my bike in a repair stand where it hangs by the seatpost. I was wondering if this might not be ideal and that the prolonged hanging might put too much stress on the bike and that I should instead store it standing on the floor. Does this way of storing the bike pose any chance of damaging the bike compared to storing it standing?
I don't think that this should be a problem. The force that is transferred through the seatpost to the bike when you sit on it and ride some bumpy road is vastly more than the stress that your bike might get from hanging on the bike stand.
It might even be better than standing it on the same patch of tyres, if it's not being ridden at all, especially if you're not checking the tyre pressures.
Nope, not a problem.
The stress on non-moving components comes from repeated cycles of loading and unloading, not from static force. As others have pointed out, keeping the tires unweighted might help lengthen their life in storage, though it should make no difference with the bearings since they're metal on metal with no movement.
The only thing you'd need to watch out for is over tightening the clamp around the seatpost, although this has nothing to do with the amount of time that it sits. If you clamp it too tight the damage is done in an instant.