Extradimensional space = planar travel
Extradimensional spaces are explicitly listed in the section talking about planes. That means travelling to or from one is cross-plane. Consequently, using the Bottled Respite ability from the Genie's Vessel feature involves planar travel, and thus can't be done inside an active Private Sanctum that's been configured to block it.
If MPS is cast outside: Leaving the Respite may be tricky
For the Genie-patron warlock, the description of the Bottled Respite ability of the Genie's Vessel feature says (TCoE, p. 73-74):
You can remain inside the vessel up to a number of hours equal to twice your proficiency bonus. [...] When you exit the vessel, you appear in the unoccupied space closest to it.
So if you're in the vessel, then someone bars planar travel with a Private Sanctum that will last long enough to bar leaving the bottle when the duration expires, what happens?
Technically, the rules don't say. Delaying leaving the bottle explicitly contradicts one of the Bottled Respite ability's rules, though, and there's another way to do it that doesn't. I would treat the whole area of the Private Sanctum as occupied space (because you can't materialize there). When the effect expires, the Respite's contents reappear in the closest unoccupied space outside the Sanctum.
If MPS is cast inside: Requires judgment call
Leaving the space is definitely planar travel. In fact, the question of how to define the area of effect of Mordenkainen's Private Sanctum is rather interesting, as it can be specified to cover an area large than the interior of the vessel, and what happens to the 'excess volume' isn't addressed.
But the spell easily covers the entire demiplane. The spell's description therefore directly conflicts with the class feature's. They can't be reconciled. One of them also isn't clearly 'more specific' than the other. One is specific to a particular spell, while the other is specific to a particular item related to a particular class feature.
Either the spell wins (forcing additional duration on the Bottled Respite ability), or Bottled Respite's rules win (forcing an exception to the spell). There really isn't any way around it, even by trying to split the difference (if you say failure to exit the Vessel on time destroys it, that still results in planar travel).
I honestly don't see any logical reason to prefer one superseding the other. It will come down to a judgment call.
There is one way to resolve this that doesn't technically violate either rule, but I don't like it (as it's a pretty unpleasant 'gotcha' along the lines of old-school teleport errors). When the Private Sanctum spell prevents Bottled Respite from ending normally, it destroys the vessel. The extradimensional space having been destroyed, the Private Sanctum spell no longer targets a valid target; therefore, it ends simultaneously with the contents being ejected normally.
The ability text for the Vessel covers its destruction
A later part of the description of Bottled Respite adds:
[...] if the vessel is destroyed, every object stored there harmlessly appears in the unoccupied spaces closest to the vessel's former space.