ag.algebraic geometry - What does being Analytically Isomorphic imply for classification of singularities on curves?

Analytically equivalent (over C) implies, for instance, topologically equivalent. All information items Charles Siegel mentions (number of branches, tangencies, etc, and also the classification of double points) are topological invariants so in particular they are analytic invariants and can be read off the completion indeed.

I would say that the first step to classify singularities is equisingularity, which over C is the same as topological classification. The analytic classification of is extremely complicated, even for plane curves. For each topological class there is an analytic moduli space, but it need not be irreducible, equidimensional, or even separated.

For plane curve singularities, there are a few cases where analytically equivalent and topologically equivalent are the same thing. These are the so-called simple, or Du Val, singularities, namely "A-singularities" or double points, whose equations have the form $y²-x^n=0$, "D-singularities" or $x(y^2-x^n)=0$ and $E_6:y^3-x^4=0$, $E_7:y(y^2-x^3)=0$, $E_8:y^3-x^5=0$.

A good old reference for the analytic classification of plane curves is Zariski's booklet ''Le problème des modules pour les branches planes.'' As introduction to the theory of plane curve singularities you have Casas-Alvero ''Singularities of plane curves'', and Laudal-Pfister's LNM ''Local moduli and singularities'' is also recommendable.