The location of an object x depends on how we choose our coordinate system.
This is not true. The position of an object is invariant; it is the label which we assign to that position which depends on our coordinate system.
Take out a blank sheet of paper and draw a dot on it. Where is the dot? It seems like a bit of a non-answer, but a reasonable response would be "it is where it is."
This isn't particularly useful for computing things, so we typically decide to assign the point a numerical label. We can do this in a number of ways - we could choose a rectilinear grid, or we could choose a polar grid, or something more exotic; we could choose the coordinate origin to be in the center of the page, or we could choose the bottom left instead; and given any valid coordinate system, we can stretch or compress it to get a different one.
However, the dot that you drew on the paper hasn't moved - swapping labels around, which physicists do in our imaginations, has no effect on the world, or on any measurements we could perform in it. A coordinate system is simply a choice of labels for the observable quantity position.
How are these two situations different?
At this level, they aren't. Saying that the coordinates of a particular point are $(2,3)$ means absolutely nothing unless I specify a coordinate system, which essentially amounts to a choice of gauge. Similarly, saying that the global phase of a wavefunction is $\pi/4$ is meaningless unless I establish a reference point of some kind. This wouldn't be impossible - I could demand that the wave function evaluated at $x=0$ be purely real at $t=0$, and then calculate the global phase of the wave function at any other time based on this reference point.
The difference lies in the fact that position is an observable quantity while the wave function is not. If it were somehow possible to ascertain the precise value of $\psi(x)$, then we could set a reference point as mentioned above and define a meaningful notion of observable global phase. However, as we can only actually measure $|\psi|^2$, we can't go backward to unambiguously determine $\psi$, and so the global phase of a wave function does not have measurable physical content.