2 Weakened the assumption of non-singularity.; added 18 characters in body

Yes, assuming at least that the cone point is the only singular point. Hence any isomorphism will preserve the ideal of that point which is the ideal of elements of positive degree.(I think that the cone should in some sense be the most singular point in general and hence would still be preserved.) You can think of the isomorphism and its inverse as a graded isomorphism where the coordinate rings are graded by the powers of the ideal of the cone point. They then give graded isomorphisms of the associated graded rings. These associated graded rings are however the original coordinate rings. Hence we get a graded isomorphism of the coordinat rings and these isomorphisms are equal to those induced by the linear maps on the degree $1$ part.

Addendum: You can considerably weaken the condition that the cone point is the only singular point. Assume that the associated projective scheme to he ideals are varieties which are not cones. Then the multiplicity of any point outside of the cone point is equal to the multiplicity of the image point on the projective variety and that multiplicity is smaller than the degree of the variety. That degree however is just the multiplicity of the cone point. Hence, the cone points are the points of maximum multiplicity on the respective variety and are therefore taken to each other by an isomorphism. It seems likely that the case of the projective varieties being cones can be further analysed.

1

Yes, assuming at least that the cone point is the only singular point. Hence any isomorphism will preserve the ideal of that point which is the ideal of elements of positive degree.(I think that the cone should in some sense be the most singular point in general and hence would still be preserved.) You can think of the isomorphism and its inverse as a graded isomorphism where the coordinate rings are graded by the powers of the ideal of the cone point. They then give graded isomorphisms of the associated graded rings. These associated graded rings are however the original coordinate rings. Hence we get a graded isomorphism of the coordinat rings and these isomorphisms are equal to those induced by the linear maps on the degree $1$ part.