Correct answer is yes, especially in the form of:
- So, at same height relative to sea level, would the atmospheric pressure be the same?
As there is no difference between isobar surface formed by liquid or gas, it will be the same surface.
One of the reasons for strict yes is that because if not, then we will have flow of mass of liquid or gas, as result of what pressure is(kinetic movement of molecules in essence) and it allows us to have perpetuum mobile or break second thermodynamic law, or conservation of energy or mass - which obiviously won't happen by magic of donout shaping planet size amounts of mass - magic is not strong enough in this one.
It is not an answer to q, or it is a half baked answer, because the actual shape of that equipotencial surface should be part of such answer, maybe, but I'm not prepared to answer that part, and this answer is more to counterweigth JBH present answer. Sure there are fluctuations of pressure on earth, for a number of reasons (not that many reasons btw), but pressure deviations are just that deviation, and it seems those deviations are not the subject of the q, at least not before first order of the problems is shown/solved.
PS I can't comment cuz some funky changes to wb javascrips, so it is - answer comment.
There is such work The exterior gravitational potential of toroids, unfortunatly no nice pictures for us to easy understand things, except fig2. Probably okay work, but for pictures for the absence of general view picture it 2 points out of 10. But formulas there, so potencially gourging one to plug in some plot producing thing is here.
This work is better, to some degree, fig8 in it, Gravitational potential of a homogeneous circular torus: new approach