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Your polar coordinate system still needs an “angle 0” which is somewhat analogous to “North”.
The sun would be the most obvious choice for such a reference point, though it'd be amusing to make it Greenwich and therefore make everyone deal with Earth's rotation and orbit if they want extremely precise calculations (though I expect there wouldn't really be a practical difference on that scale).
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.