Trying to generalize Torsten's answer: It seems that if the cones are isomorphic then the isomorphism can indeed be chosen to preserve the origin.

For a projective variety $V$ let's denote the affine cone by $C(V)$. Torsten says that if $V$ is not a (projective) cone then in $C(V)$ the "cone point" $0$ is the unique point of maximum multiplicity.

$V$ is a projective cone if it is the join of a point $\mathbb P^0$ in the ambient projective space with a variety $W$ in a hyperplane (a hyperplane not containing that point). In this case $C(V)$ is the product of $C(\mathbb P^0)=\mathbb A^1$ and $C(W)$. In the general case $V$ is the join of a linear $\mathbb P^{d-1}$ with some $W$ which is not itself a projective cone, and then $C(V)=C(\mathbb P^{d-1})\times C(W)=\mathbb A^d\times C(W)$. Surely Torsten's statement generalizes to say that the points of maximal multiplicity in $C(V)$ are now those in $\mathbb A^d\times 0$.

So, given $V_1$ and $V_2$ such that $C(V_1)$ and $C(V_2)$ are isomorphic, the two numbers $d_1$ and $d_2$ must be equal, and if the isomorphism does not carry $0$ to $0$ then it can be adjusted to do so using translations in $\mathbb A^d$.

EDIT in response to Orr's comment: Here's what I mean, in your notation. Let $I$ be a homogeneous ideal in $k[x_1,\dots ,x_n]$ corresponding to some "homogeneous" affine variety $V(I)\subset\mathbb A^n$ and let $P(I)\subset \mathbb P^{n-1}$ be the variety that the same ideal defines. (I believe $V(I)$ is called the affine cone on $P(I)$.)

It might happen that after some linear change of coordinates $I$ becomes an ideal generated by a homogeneous ideal $I_1$ in $k[x_1,\dots ,x_p]$ and a homogeneous ideal $I_2$ in $k[x_{p+1},\dots ,x_n]$. If so, then $V(I)$ is the product of $V(I_1)\subset \mathbb A^p$ and $V(I_2)\subset \mathbb A^{n-p}$, and I believe that $P(I)$ would be called the join of the projective varieties $P(I_1)\subset \mathbb P^{p-1}$ and $P(I_2)\subset \mathbb P^{n-p-1}$. In particular if $p=n-1$ and $I_2=0$ then $V(I)=V(I_1)\times \mathbb A^1$ and $P(I)$ is called the projective cone on $P(I_1)$.

Torsten is arguing that if $P(I)$ is not a cone, i.e. if there is no linear change of variable such $I$ is generated by polynomials not involving the last coordinate, then the origin is intrinsically characterized as the unique point in $V(I)$ of maximal multiplicity. I am saying that one can treat the general case in the same way, as follows: Suppose that $P(I)$ is a cone, or a cone on a cone, or ... as far as you can go. That is, make a linear change of variables so that $I$ is generated by polynomials in the first $p$ coordinates with $p$ as small as possible. Thus $V(I)$ is the product of some $V(I_1)$ with $\mathbb A^{n-p}$ and $P(I)$ is the join of the corresponding $P(I_1)$ with $\mathbb P^{n-p-1}$. Now in $V(I)=V(I_1)\times \mathbb A^{n-p}$ the points of $0\times \mathbb A^{n-p}$ are the points of maximum multiplicity, and furthermore any one of them is carried to $0=(0,0)$ by some automorphism of $V(I)$ since $\mathbb A^{n-p}$ has an automorphism group that acts transitively.

The idea of iterated singular locus is not quite so successful. In most cases if the projective variety $S(P(I))$ is $P(J)$ then the homogeneous affine variety $S(V(I))$ will be $V(J)$. In the extreme case when $P(I)$ is smooth, so that $S(P(I))$ is empty, $S(V(I))$ will be $0$, with the exception that if $P(I)$ is a projective space (linearly embedded in $\mathbb P^{n-1}$) then $V(I)$ will be an affine space (linearly embedded and containing the origin) whose singular locus is empty rather than $0$. Thus in the sequence $V(I)$, $S(V(I))$, ... the last nonempty thing will be an affine space, possibly $0$ or possibly bigger. But it will not always be the same thing as before (the maximal affine space such that $V(I)$ is in a linear fashion the product of it with something). For example, if $n=3$ and $P(I)$ is a projective curve which has exactly one singular point but which is not simply the union of lines through that point, then the singular locus of the homogeneous surface $V(I)\subset A^3$ will be a line through the origin but there will be no automorphism of $V(I)$ moving $0$ to another point in that line (or to anything other than $0$).