Coordinate system vs ordered basis

(1) Given any finite-dimensional vector space $\Bbb V$, a choice $(E_a)$ of basis determines a linear isomorphism $\Phi : \Bbb R^n \to \Bbb V$, $n := \dim \Bbb V$, by $$\Phi(x^1, \ldots, x^n) := x^a E_a .$$ The inverse map, $\phi := \Phi^{-1} : \Bbb V \to \Bbb R^n$ defines a preferred global chart on $\Bbb V$ and so realizes $\Bbb V$ as a smooth $n$-manifold. (Pace the claim in the question, this procedure does define coordinates on $\Bbb V$ in the differential-geometric sense.) We might call these charts $\phi$ linear coordinate charts on $\Bbb V$.

In turn, this choice determines a global frame of the tangent bundle, $T\Bbb V$, namely, $(\partial_{x^a})$, which in turn restricts at each point $v \in \Bbb V$ to a basis $(\partial_{x^a}\vert_v)$ of $T_v \Bbb V$. What's special to the case of a vector space is that for each $v \in \Bbb V$ there a canonical identification $\Psi_v : T_v \Bbb V \to \Bbb V$, namely, $$\Psi_v : v^a \partial_{x^a}\vert_v \mapsto v^a E_a,$$ or just $$\Psi_v : V \mapsto dx^a(V) E_a .$$ Here canonical means that this identification doesn't depend on our choice of basis, it's a natural, built-in feature of $\Bbb V$. Note, by that way, that when $n > 0$, not all coordinate charts on $\Bbb V$ arise from a basis---there is only an $n^2$-parameter family of bases, but uncountably many choices of coordinate charts.

(2) Now, we can see that for any $v \in \Bbb V$, any basis $(F_a)$ of $T_p \Bbb V$ is the restriction of a global frame induced by a choice of basis of $\Bbb V$, namely, $(\Psi_v(F_a))$. More generally, given any smooth manifold $M$, point $p \in M$, and basis $(F_a)$ of $T_p M$, one can construct a smooth chart $(U, \varphi)$ on $M$, $U \ni p$, such that $F_a$ is the restriction of the coordinate frame, that is, such that $T_0 \varphi \cdot F_a = \partial_{x^a}\vert_0$.

On the other hand, not every local frame of $\Bbb V$ is induced by a choice of basis of $\Bbb V$---and indeed, not every local frame on a smooth manifold $M$ is not a coordinate frame! Thus, for our vector space $\Bbb V$ (when $n > 1$) there are proper inclusions

$$ \begin{align} &\{\textrm{coordinate frames of linear coordinate charts on $\Bbb V$}\} \\ &\qquad\qquad \subsetneq \{\textrm{(local) coordinate frames on $\Bbb V$}\} \\ &\qquad\qquad\qquad\qquad \subsetneq \{\textrm{(local) frames on $\Bbb V$}\} \\ \end{align} $$

One way to see that the latter inclusion holds (for all manifolds of dimension $n > 1$) is to define the Lie bracket operation $[\,\cdot\, , \,\cdot\,] : \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)$. In coordinates it is given by $$[X, Y] := (X^b \partial_{x^b} Y^a - Y^b \partial_{x^b} X^a) \partial_{x^a} ,$$ but computing the transformation of $[X, Y]$ under a general change of coordinates shows that it does not depend on a choice of coordinates. For any coordinate chart $(U, \phi)$, the above formula gives that the coordinate frame $(\partial_{x^a})$ satisfies $[\partial_{x^a}, \partial_{x^b}] = 0$ for all $a, b$. On the other hand, most local frames $(E_a)$ do not satisfy $[E_a, E_b] = 0$ and so cannot be the coordinate frames of any coordinate chart.