Is $f(x,y) = f(\mathbf{x})$ abuse of notation?

From page 32 of Halmos's book, Naive Set Theory:

$f(\mathbf{x}) = f((x, y))$ is more commonly written as $f(x, y)$ (where $\mathbf{x}=(x, y)$ is a vector in e.g. $\mathbb{R}^2$). The same thing happens with longer tuples: $f(x_1, \ldots, x_n)$. Pedantically speaking, $f((x, y))$ is indeed the (more) correct notation. I suppose the reason for omitting the extra parenthesis (besides convention) is that they are simply reduntant; removing them causes no ambiguity whatsoever.

I would not call this abuse of notation as we are just using two different notations for the same thing.

We are basically just switching back and forth between vectors $\vec x = xi+yj $ and the coordinates of the vectors $(x,y)$ with respect to a certain basis $(i,j)$. (Usually the basis is implicitly the cannonical one, or can be derived from the context.) As the coordinates of a vector with respect to a given basis are unique, this is does not allow any ambiguity.

Your example is indeed correct, and it obviously depends on how whether you define first a vectorial derivative or first the partial derivative. In any case, the derivative in the direction of a basis vector is equal to the derivative with respect to the corresponding coordinate.