Classically, if $X \subseteq \mathbf{R}^{3}$ is an open set, then a $3$-dimensional coordinate system on $X$ is nothing but an injective, continuously-differentiable mapping $\xi:X \to \mathbf{R}^{3}$ whose differential has rank three at each point. Conceptually, a coordinate system assigns an ordered triple of real numbers to each point of $X$ in such a way that distinct points are given distinct coordinates, satisfying some technical conditions of smoothness. The inverse mapping $\xi^{-1}:\xi(X) \to X$ is sometimes called a parametrization of $X$.
The inclusion map $i:X \hookrightarrow \mathbf{R}^{3}$ is a coordinate system on $X$. If $A$ is an invertible $3 \times 3$ real matrix and $b$ is a vector in $\mathbf{R}^{3}$, then the affine map $T(x) = Ax + b$ defines a coordinate system on $X$. Cylindrical and spherical coordinates are parametrizations of portions of $\mathbf{R}^{3}$; for example, the spherical coordinates mapping (with geographic angles)
$$
S(r, \theta, \phi) = (r\cos\theta\cos\phi, r\sin\theta\cos\phi, r\sin\phi),\qquad
0 < r,\quad |\theta| < \pi,\quad |\phi| < \pi/2
$$
parametrizes $X = \mathbf{R}^{3}\setminus \{y = 0, x \leq 0\}$, the complement of a closed half-plane. The spherical coordinates of a point $(x, y, z)$ of $X$ are the numbers $(r, \theta, \phi)$ such that $(x, y, z) = S(r, \theta, \phi)$.
In linear algebra (over an arbitrary field $F$), a "coordinate system" for an $n$-dimensional vector space $V$ takes values in $F^n$, and is usually defined by choosing a basis $B = \{v_{i}\}_{i=1}^{n}$ and mapping a vector $v$ in $V$ to its coordinate vector $[v]_{B}$ in $F^n$. Similarly, in affine geometry one can construct a coordinate system as you describe (by picking an origin and a basis for the resulting vector space).
In the modern point of view, coordinate systems play a secondary role to "overlap maps". For concreteness, let $X$ be an open subset of $\mathbf{R}^{3}$. First, we weaken the criteria for a mapping to be a coordinate system, requiring only that $\xi:X \to \mathbf{R}^{3}$ be continuous and injective. Now assume $\xi_{1}$ and $\xi_{2}$ are "allowable" coordinate systems on $X$ (for some value of "allowable", yet to be determined). An overlap map is a composition
$$
\xi_{1} \circ \xi_{2}^{-1}:\xi_{2}(X) \to \xi_{1}(X).
$$
The "structure" of $X$ is encoded in the properties of the overlap maps. For example, if the overlap maps are diffeomorphisms, then "smoothness" makes sense for functions $f:X \to \mathbf{R}$: We pick an arbitrary coordinate system $\xi_{1}$ and declare $f$ to be smooth if the composition $f \circ \xi_{1}^{-1}$ is smooth as a function on $\mathbf{R}^{3}$. This definition does not depend on $\xi$, since
$$
f \circ \xi_{2}^{-1} = (f \circ \xi_{1}^{-1}) \circ (\xi_{1} \circ \xi_{2}^{-1}),
$$
and the overlap map is a diffeomorphism. Similarly, if the overlap maps are affine, then (for example) "line segments" in $X$ make sense: A subset $\ell$ of $X$ is a "segment" if in some (hence every) coordinate system $\xi$, the set $\xi(\ell)$ is a line segment.
Philosophically, a coordinate system is merely how one "transfers" objects and functions on a space $X$ to a objects and functions on a "standard" space, such as $\mathbf{R}^{3}$. The interesting "structure" of $X$ is encoded by the overlap maps, which determine properties of $X$ that are independent of the coordinate system.
When mathematicians speak of a manifold having a smooth structure, they mean that some collection of coordinate systems has been fixed so that the overlap maps are diffeomorphisms. A manifold having an affine structure has coordinate systems whose overlaps are affine (a much more stringent condition). Similarly, one hears of holomorphic, piecewise-linear, and conformal structures, among many others.
To address the question about coordinate vector fields in spherical coordinates: Though the spherical coordinates map parametrizes part of $\mathbf{R}^{3}$ (which is a vector space), the spherical coordinates mapping is not a linear transformation, and therefore does not belong to linear algebra, but instead to multivariable calculus.
The "proper framework" requires some explanation. If $X \subseteq \mathbf{R}^{3}$ is an open set, define the tangent bundle $TX$ to be $X \times \mathbf{R}^{3}$. An element $(\mathbf{x}, \mathbf{v})$ of $TX$ should be viewed as consisting of an element $\mathbf{x}$ of $X$ together with a vector $\mathbf{v}$ "based at" $\mathbf{x}$. It makes sense to take linear combinations of vectors only if they're based at the same point.
In this picture, a vector field on $X$ is a mapping $\Xi:X \to TX$ that assigns, to each point $\mathbf{x}$ of $X$, a vector $\Xi(\mathbf{x})$ based at $\mathbf{x}$. The Cartesian coordinate fields $\mathbf{e}_{i}$ are constant vector-valued functions because Cartesian coordinates change by additive constants under translation (!). By contrast, the spherical coordinate fields are non-constant, because spherical coordinates do not change in a simple way under translation.
This is not a proper answer, but it might be useful anyway. It's just what I've found so far reading books and trying to make sense of everything I learned about vectors in both calculus and algebra.
You can geometrically picture vectors in $\mathbb{R}^n$ as arrows placed at the origin. Every vector can be uniquely expressed as a linear combination of $n$ linearly independent vectors, so for every basis of this vector space $(\vec{e}_1,\dots,\vec{e}_n)$ we can write:
$$
v=v^1 \vec{e}_1 + \cdots + v^n \vec{e}_n
$$
Although using a vector basis allows us to uniquely identify every vector in $\mathbb{R}^n$, it isn't really useful when trying to identify every "point" in $\mathbb{R}^n$ because of its limitations (your axes will have to be straight lines and will have to include the "canonical" origin).
In order to identify every point of $\mathbb{R}^n$ with more freedom, our first approach could be affine geometry. In $\mathbb{R}^n$ viewed as an affine space, we can define a coordinate system $(O,\mathcal{B})$, with $O$ a point in $\mathbb{R}^n$ and $\mathcal{B}$ a basis of $\mathbb{R}^n$ as a vector space. Axes are still straight lines, but now we can move from the origin to any other point $O$. This affine coordinate system is in a way a coordinate system, and it is definitely not the same as a basis — because $(O,\mathcal{B}) \neq \mathcal{B}$ — but we can do better.
We can define a system of $n$ equations (not necessarily linear, a system of linear equations would bring us back to the affine case) that uniquely identify every point in $\mathbb{R}^n$:
$$
x^i =\Phi^i(q^1,\dots,q^n), \space\space\space i=1,\dots,n
$$
This is precisely what we do when we define cylindrical or spherical coordinates: express $x,y,z$ in terms of three new variables ($\rho,\varphi,z$ and $r,\theta,\varphi$ respectively).
This means that the values $(q^1,\dots,q^n)$ will be our new coordinates. Note that, if this system were linear, we would need to require the coefficient matrix to have a non-zero determinant in order for this system to have a (unique) solution. For a general system of equations, by virtue of the implicit function theorem, the analogous condition is:
$$
\frac{\partial(x^1,\dots,x^n)}{\partial(q^1,\dots,q^n)} \neq 0
$$
i.e. the Jacobian of the transformation must be non-zero.
This new coordinates $(q^1,\dots,q^n)$ aren't related to any basis, but they induce one for every point $(q^1,\dots,q^n)$ in $\mathbb{R}^n$: the so-called coordinate basis of this coordinate system:
$$
\vec{v}_\mu = \frac{\partial\vec{\Phi}}{\partial q^\mu}
$$
In this sense, we can see than the components of the position vector at the point $p\in\mathbb{R}^n$ will be different from the coordinates of the point $p$ itself. The components depend on the coordinate basis (or any other basis which you define in terms of that one), while the coordinates of a point depend on the coordinate system itself.
In fact, now we're not talking about $\mathbb{R}^n$ as a vector space anymore, but this space does have a vector space attatched at every point, a vector space we call the tangent space at $p$: $T_p\mathbb{R}^n$. The coordinate basis at every point is the vector basis that our coordinate system induces for the tangent space at that point.
Note that the components of a vector (or a tensor, for that matter) may be called coordinates. I only see this when reading about pure vector spaces, without any sense of geometry, metrics or anything. For example, a matrix $A\in\mathcal{M}_{n\times n}$ which satisfies $\vec{v}_{\mathcal{B}'} = A\vec{v}_{\mathcal{B}}$ might be called a change of coordinates matrix (from the coordinates from $\mathcal{B}$ to the coordinates from $\mathcal{B}'$) or a change of basis matrix (from the basis $\mathcal{B}'$ to the basis $\mathcal{B}$, because it satisfies $\mathcal{B}'A=\mathcal{B}$, if we allow ourselves this abuse of notation).
Nonetheless, I always say components when referring to a vector (or tensor), and coordinates when referring to a point.
Best Answer
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.