Confusion with vector spaces

The vector space axioms tell how addition and scalar multiplication in a vector space must behave. The Euclidean plane with addition and scalar multiplication defined using the usual coordinate system satisfies those axioms. With those definitions the arithmetic agrees with your intuition about how vectors should behave geometrically.

Those axioms were chosen to capture the idea of "linearity".

There is no reason to expect that arbitrary other coordinate systems will allow you to calculate vector sums one coordinate at a time. In fact, the ones that do are precisely those defined by applying an invertible linear transformation. In vector space terms that's choosing a different basis.

You got a good answer already, but here's another way of putting it.

If two vectors, say $v$ and $w$, are a basis for $\mathbb R^2$ (for example, vectors $\hat i$ and $\hat j$),

then any vectors $a$ and $b$ in $\mathbb R^2$ can be expressed as a linear combination of $v$ and $w$,

say $a=a_vv+a_ww$ and $b=b_vv+b_ww$. Because of the properties of vector spaces,

then $a+b=a_vv+a_ww+b_vv+b_w=(a_v+b_v)v+(a_w+b_w)w$,

so we add vectors component-wise.

On the other hand, when we express vectors using polar coordinates $r$ and $\theta$,

the vectors are not expressed as linear combinations of two vectors,

but rather as $(r\cos\theta,r\sin\theta)$, which is not linear in $\theta$,

so addition of vectors is not so convenient in polar coordinates,

though other operations may be more convenient in polar coordinates.