The phenomenon you are talking about is essentially that of a homomorphism, or homomorphic structures. That is, structures that appear superficially different but share an underlying common structure.

The concept of a 'functor' was invented to describe a higher order 'homomorphism of homomorphisms'. An example most people miss is the total derivative in multivariable calculus: the chain rule implies that the total derivative is a functor that maps the composition of differentiable functions on a manifold, to matrix multiplication (of matrices acting on the tangent space).

You might also be interested in various 'dual' concepts, like that between tangent spaces and cotangent spaces in differential geometry.

For algebra, I'd recommend Pinter's Book of Abstract Algebra.