The concept of a coordinate system is not ambiguous. You have dimensions, and each can be represented as a vector that complies with specific properties, such as linear independence.
> Which cardinal direction map to x, y, z respectively (e.g. is z for up/down or forwards/backwards)?
That's a function of whatever coordinate transformation you wish to apply.
Nevertheless, I vaguely recall from school the concept of an oriented vector space and direct coordinate system, whose definition was something like the cross product of consecutive director vectors resulted in a positive vector (right-hand rule) i.e., the direction of z is determined unambiguously by the direction of x and y.
> Now in geometric algebra we also have oriented basis vectors.
If I'm not mistaken, oriented referential systems are covered in intro to euclidean geometry classes.
> The key difference is that geometric algebra has the exterior product, notated ^. For example, e1 ^ e2 is the exterior product of two oriented basis vectors. You can interpret this as being an oriented basis for the space spanned by e1, e2. And similarly for e1 ^ e2 ^ e3 being an oriented basis of the volume spanned by e1, e2, e3, etc. These are called basis blades.
Sounds like a convoluted way to refer to basic concepts like direction vectors of a direct coordinate system.
Quite bluntly, this all sounds like an attempt to reinvent euclidean geometry following a convoluted way. I mean, what does all this buy you that applying a subset of affine transformations (scaling, translation, rotation) to an orthogonal coordinate system doesn't give you already?