Make one object the same shape as another?

What software are you using for this? Most 3d software will have the ability to morph an object. If you have two of the same exact mesh, one having been stretched or translated into a shape but the other is the base shape, then that's a perfect way to use morphs to transform the base shape into the distorted shape automatically without the need to repeat the modeling steps. In some software apps its called morphs, morpher, or morphing. In some others it's called blend shapes or blend keys. Depends on your software. The key to using this process is that the deformations must not affect the number of vertices on the object. The vertex order must be the same between the base shape and the distorted shape.

I'm going to say yes lol because without knowing what your software is it is impossible for anyone to give you a firm answer.

Another solution would be to use conforming or retopology tools to "wrap" the base mesh to the surface of the deformed mesh. Unlike a morph, this process will work even if the vertices do not match.