My query is a little vague, but I'll try to be as concrete as possible.
Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on the leading-order terms of the defining equations at that point?
I will elaborate a little here:
Geometric intuition suggests that the 'structures' of $z^2 - xy = 0$ and $x^4 + z^2 - xy = 0$ are 'the same' at the origin, because they differ only by a higher-order term $x^4$. On the other hand, $z^2 - xy = 0$ and, say, $z^2 - xy + x^3 + y^3 + z^3 = 0$ seem to be different algebraically, as in the first case we don't have unique factorisation in the local ring at the origin ($z^2 = xy$), whereas in the second I think we do (although perhaps this is wrong).
I am aware that the answer may depend on the category in which we work (algebraic vs. analytic).
In terms of classifying singularities, I usually think in terms of the ideal generated by the partial derivatives of the defining equations of the variety and also the defining equations themselves. You can think just in terms of ideals if you like for classifying singularities, but it's the associated scheme structure on the singularity corresponding to the ideal mentioned above which gives the singularity a more geometric structure. This scheme structure on the singularity is referred to as the singular subscheme of a singular variety. For the two examples you give the ideals corresponding to the singular subschemes are $(-y,-x,2z,z^2-xy)$ and $(4x^3-y,-x,2z,x^4+z^2-xy)$ respectively, which look to be the same ideal at first glance, thus their singularities have the same structure (at least from this point of view, and they have the same tangent cones as well). I'm not sure if this helps with your question much but this is how I like to distinguish between singularities.