Within the Milky Way a polar (cylindrical) coordinate system makes more sense than Cartesian - there’s an axis of rotation to define the center and ‘up/down’ directions. Zero degrees is arbitrary but a line from the galactic center to Sol, projected onto the galactic plane, would be an obvious choice as a sort of galactic prime meridian. ‘North’ and ‘south’ don’t really map to a roughly disc shaped galaxy - you’d use distance from center, angle, and ‘elevation’.
On an intergalactic scale, the center of our own galaxy is probably still the obvious choice for a center point. We could use the same axis and meridian - I don’t think the rotation of our galaxy matters on any human timescale, and on the time scales where it does matter, everything is moving relative to each other so coordinates already aren’t ‘fixed’. I’d use a spherical coordinate system instead of cylindrical for intergalactic coordinates, since things are not roughly in a plane anymore.
If you want a fixed coordinate you’d have to include a time dimension, and as the zero point for time I propose the Unix epoch. Not because it makes any sense but because it’s extremely funny to imagine computer systems in the year 10000 still relying on that legacy decision. Though special relativity makes ‘point in time’ rather complex as well - I don’t know enough to know what you’d actually need to make that work.
Of course we already have such coordinate systems for astronomy if you want to know the ‘real’ answer, one of them is pretty close to what I just came up with: https://en.wikipedia.org/wiki/Astronomical_coordinate_systems
Somewhat but not exactly. You can only go north or south until you hit the poles, but you can travel forever at angle zero starting from the center point. I guess you could call towards the center point “North” and away from it “south”, so the galactic center is the North Pole but there is no equivalent South Pole. But angle zero is more analogous to the prime meridian - it’s a line that goes north-south but there are an infinite number of such lines, and we could have called any of them zero.