Using a Cartesian basis set allows you to unambiguously identify any point in 3 space with the fewest coordinates.

Try it using some other value (say, 60°, in 2D) and you'll see that you need three axes, a needless complication when 2 would suffice.

You'll find that 3D rotations become excessively complex - a 2D non-Cartesian system is one thing - but in 3D there must be a preferred axis of symmetry...

(I'm imagining a duodecahedron, spun between thumb and forefinger)

...and that rather goes against the spirit of keeping the maths simple.

In addition to James answer, any arbitrary non-zero angles between axis (see Skew coordinates from wiki) don't change the distances and the corresponding two metric spaces (Cartesian and Skew) are identical (see isometry). With regards to observable universe we deal with spacetime metric. 

I guess your question is why the basis used to describe a space is composed of vectors with direction 90° appart?

Actually it is not necessary, you could describe any position of a point in 2D with 2 vectors 60° appart. They must just not be colinear. It is also possible in any dimension. But as pointed out by J. Garry, it is not as simple as using an orthogonal basis. For example, if you have a 3D {x,y,z} basis and you decide to dilate the x direction, y and z remains unchanged if they are orthogonal to x, not otherwise.

Note that for non euclidian spaces (such as in general relativity), the orthogonality of the basis is not globaly ensured (the angle is not always 90°), and this leads to awfully complicated equations.

Nonetheless, some fields in physics uses non-cartesian basis, such as cristallography (as far as I remember), because the symmetry of the studied objects may be better described otherwise.

So, the answer seems to be based on convention and easy maths. Imagine also the possibility of a vectorless space in the universe, relative to the coordinates (0,0,0,). That is, one cannot define a vector relationship in certain parts of the universe, because of hidden dimensions that disallow such connectivity. This appears to be the case perhaps, in quantum physics. 

It might not be necessary in the Cartesian Coordinate system alone. But when it comes to conversion of the Cartesian coordinates to Polar (Cylindrical and Spherical) Coordinate Systems it is really difficulty for us to determine the unique position of any point in space when the axes are not perpendicular to each other. We practice with the axes which are 90 degree to each other and that have made it easier to convert the coordinate system through the use of Pythagorean Equation. Otherwise it would have been difficult and vague.