The rules on extradimensional spaces are called out in the text, as the sanctum is similar to a mage's magnificent mansion, and thus inherits the extradimensional quality. There are five points of interest, I think.
"Extradimensional Spaces: A number of spells and magic items utilize extradimensional spaces, such as rope trick, a bag of holding, a handy haversack, and a portable hole. These spells and magic items create a tiny pocket space that does not exist in any dimension."
"Demiplanes: This catchall category covers all extradimensional spaces that function like planes but have measurable size and limited access."
"Mage's Magnificent Mansion: Since the place can be entered only through its special portal, outside conditions do not affect the mansion, nor do conditions inside it pass to the plane beyond."
"Mage's Magnificent Mansion (Effect): extra-dimensional mansion, up to three 10-ft. cubes/level (S)"
"Create Lesser Demiplane (Effect): extradimensional demiplane, up to three 10-ft. cubes/level (S)"
The question is whether or not a mage's magnificent mansion counts as an extradimensional space that functions like a plane (note that the effect of the spell is not the same as spells that create a demiplane, creating an extradimensional mansion rather than an extradimensional demiplane). Additionally, whether or not the sanctum inherits the quality of the magnificent mansion that the place can be entered only through its special portal (not entirely, since sanctum calls out that the familiar can enter as a full-round action and modifies the conjuring of the door, but all other factors should be in place), and that the mansion is distinguished as a different extradimensional space to the plane beyond, meaning it isn't a plane.
If you decide that the magnificent mansion is an extradimensional space that does function like a plane (an extradimensional cordoned-off section of a plane), then yes, you can plane shift or gate to it (or get there with a wish, miracle, or limited wish). If you decide that it does not function like a plane (e.g. does it have planar qualities by RAW, or is the effect of the spell creating a demiplane specifically?), then you cannot plane shift or gate to it, and only a wish or miracle would let you break in (because wish has the clause that allows it to ignore all local conditions).
I personally would rule that no, plane shift and gate do not allow you entry into a sanctum, but that wish and miracle do. That is, however, partly from the perspective that as a 6th tier mythic ability, it should be exceedingly difficult to break in (and requiring the entrant to burn a wish or miracle spell if they want to get in uninvited seems an appropriate tax), so inheriting the magnificent mansion limitation that the entry points laid out in the sanctum ability description are the only means of entry (which a wish can bypass) seems fair to me.
