In the classical Euclidean setting $\mathbb{R}^n$ with cartesian coordinates $x_i$ one can identify a vector field with a map $\mathbb{R^n}\to\mathbb{R}^n$, because every tangent space at any point can be identified with $\mathbb{R}^n$ without ambiguity.
Suppose now that one has on $\mathbb{R}^n$ (or on an open subset of it, a ball for instance) a metric different from the classical one. My question is:
up to renormalise the basis $\frac{\partial}{\partial x_i}$ with respect to the new metric, can one identify $\frac{\partial}{\partial x_i}$ with $x_i$ so that a vector field can be identified once again with a map $\mathbb{R}^n\to\mathbb{R}^n$?
The identification will be something like $b_i\frac{\partial}{\partial x^i}\mapsto b_ix^i$.
Suppose that $M$ is a differentiable $n$-manifold with trivial tangent bundle, $TM\cong M\times R^n$. For instance, $M$ can be any open subset of $R^n$.
A vector field on $M$ is just a smooth section of $TM$; in view of triviality of $TM$, it is just a smooth map $M\to TM, x\mapsto (x, v(x)), x\in M, v(x)\in R^n$. But it is the same thing as to say that you have a smooth map $x\mapsto v(x)$, i.e. a smooth map $M\to R^n$.
You do not need a metric for this, you also do not need explicit coordinates for this.