I think you are asking about real analyticity in the wrong dimension.
If you have a formal power series $\sum_{n=1} a_n x^n$ with real or complex coefficients, then you can derive the properties of the corresponding analytic function by asking its radius of convergence (which is, you know just a property of the sequence $|a_n|$).
Or, phrased differently, any real analytic function extends to a complex analytic one, and the properties of complex analytic functions are somewhat similar to real analytic functions in one dimension. If you more generally allow functions $f:\mathbb{R}\to \mathbb{C}$ to be called real analytic (with the analogous definition), then there really is no difference. The weird part there is that being complex differentiable implies analyticity.
On the other hand, a function $f:\mathbb{C}\to \mathbb{C}$ is, of course, the same as a function $f:\mathbb{R}^2\to \mathbb{C}$ and such a function could reasonably called analytic if there is a certain regular sequence of polynomials converging locally uniformly to $f$ (just like in the definition you've given).
However, these would be polynomials with complex coefficients in two variables, so the question becomes whether there exist $a_{n,k}\in \mathbb{C}$ such that
$$
f(x,y)=\sum_{n,k=0}^{\infty} a_{n,k} (x-x_0)^n(y-y_0)^k
$$
In more natural complex notation, you might write this as a power series in $z$ and $\bar{z}$ instead of $x$ and $y$ (since any two linearly independent polynomials of degree $1$ form a basis of the monomials of degree 1, and these generate the whole thing).
So what's the difference there? Well, the map $z\mapsto \bar{z}$ is an orientation-reversing involution of $\mathbb{C},$ whereas the identity clearly preserves it, so the difference between a complex analytic function and an analytic function of two variables is that the former is always the limit of orientation-preserving polynomials.