Do two circuits on one receptacle require shared trip breakers?
Yes, effectively.
Whether they are an MWBC or two separate circuits with their own neutral, same either way.  The fact that they are on the same yoke requires that they have the same shutoff.

210.7 Multiple Branch Circuits.
Where two or more branch circuits supply devices or equipment on the same yoke, a means to simultaneously disconnect the ungrounded conductors supplying those devices shall be provided at the point at which the branch circuits originate.

This concerns maintenance shutoff, not common trip.  They don't care if they trip together from an overload, GFCI or AFCI.  They care that if you plug a radio into one socket and flip breakers until the radio stops, you have de-energized the entire yoke.
You can accomplish this with listed handle ties, but since those are hard to find, we generally recommend you just use a 2-pole breaker.
However if it is wired as 2 separate circuits on 2 separate cables,  and AFCI or GFCI breakers are in use, then you must use two separate breakers with a handle tie.
If, on the other hand, they are separate yokes on the same box, then there's no special requirement.  The electrician is expected to check for that.  When putting 2 circuits in the same box, do watch your &quot;box fill&quot; (cubic inches) and use a larger box if need be.