Is there a classification of Metric Spaces?

That's too ambitious, there is an enormous variety of metric spaces. But if you restrict to specific classes, then a lot has been done. I can think of two examples: Hilbert spaces and geodesic surfaces. These are completely classified.

Hilbert spaces are vector spaces equipped with a scalar product, which induces a norm, hence a distance and so a structure of metric space, which is required to be complete. Hilbert spaces are completely characterized up to isomorphism (a stronger form of isometry); in particular, any separable real Hilbert space is isomorphic to $$ \ell^2:=\left\{\boldsymbol{x}=(x_1, x_2, x_3, \ldots)\ :\ x_j\in\mathbb R,\ \sum_{j=1}^\infty x_j^2<\infty\right\},$$ where the scalar product is given by $$ \langle \boldsymbol{x}, \boldsymbol{y}\rangle = \sum_{j=1}^\infty x_j y_j.$$ (Non-separable Hilbert spaces are classified in terms of cardinality of their orthonormal bases. Also, the same result holds in the complex case, with obvious modifications).

Geodesic surfaces are something I know less. I will refer to this beautiful note of Etienne Ghys, section 3. It is in French, but I am sure I have seen an English translation on the net.