Is the hyperbola isomorphic to the circle?
Yes, they are isomorphic.
If $u = x+yi$, and $v = x-yi$ then $$\mathbb{C}[x,y]/(x^2+y^2-1)\simeq\mathbb C[u,v]/(uv-1).$$
This can be viewed as a question in algebraic geometry but also as a question in projective geometry. Both the hyperbola and the circle are conic sections, and are projectively equivalent. In homogeneous coordinates this follows from the fact that any pair of nondegenerate indefinite quadratic forms are equivalent.