If two objects have the same gaussian curvature, are they the same up to isometries?

Intuitively, the Theorema Egregium means that you can bend or shift a space, just so long as you don't stretch it, and the Gaussian curvature will remain the same. This means that it doesn't matter how you embed a manifold in space, the curvature will always be the same. You could also interpret this as: two objects which are isometric have the same curvature.

See here for some examples of objects have the same curvature, but which are not isometric.

If two surfaces have the same Gaussian curvature, it is not true that they are the same up to an isometry.

A counterexample for that is the exponential horn ($X_1(u,v) = (u \cos v, u \sin v, \log u)$) and the cylinder ($X_2(u,v) = (u \cos v, u \sin v, v)$), which have same Gaussian curvature at corresponding points, but are actually not isometric (calculate the first fundamental form and see that they are essentially different).

What the Theorem Egregium actually says is that, if two surfaces are locally isometric, then they MUST have same Gaussian curvature at corresponding points. See it another way: if two surfaces have different Gaussian curvatures at corresponding points under a map, then that map cannot be an isometry since, by the Theorem, the surfaces would have same Gaussian curvature.