There is no real distinction to be made between extradimensional spaces and demiplanes — they're just different words for the same thing. Per the Dungeon Master's Guide, page 68, “demiplane” is just what extradimensional spaces of notable size are conventionally called. You'll also note that DMG page 99 equivocates “demiplane” and “pocket dimension”, saying that either is large enough to serve as the exotic location for a dungeon, so there isn't a real line that can be drawn between those by size, either.
So both “demiplanes” and “pocket dimensions” are extradimensional spaces, and the difference between the two is more a matter of language being vague yet evocative: the two different words just lend nuances to describing the same thing, like how “boulder” and “rock” both mean a chunk of stone, just one suggesting greater size. The difference in the words “demiplane” and “pocket dimension” is just suggesting (vaguely) something about the size of an extradimensional space.
So given that one is just a synonym for the other — yes, a familiar's pocket dimension is an extradimensional space, and could be also called a (very small) demiplane if you wanted to.
No, the extradimensional space is not attached to the wizard in any sense. If it was, there would be very clear wording about the danger of the wizard doing things like entering portable holes or rope trick spaces. That's not something that would be accidentally left out of the description of find familiar. Given that lack, it's safe for both master and familiar to enter extradimensional spaces like demiplanes and pocket dimensions. Unlike a bag of holding, there is no portal attached to the familiar (nor is there rules text about the danger) — think of it instead as the familiar being “tuned” to that personal pocket dimension, so the magic of the spell can always “find” its home pocket dimension.