Let us investigate a special type of similarity.
$$A = S(C^{-1}+D)S^{-1}$$
$$B = S(D^{-1}-C)S^{-1}$$
$$AB = S(C^{-1}D^{-1}+I - I - DC)S^{-1} \\= S((DC)^{-1}-DC)S^{-1}$$ $$BA = S(D^{-1}C^{-1}-I + I - CD)S^{-1} \\= S((CD)^{-1}-CD)S^{-1}$$
- Multiplication does "transfer" to internals and result becomes a $B$-type matrix but with $D=C$, either equal to $CD$ or $DC$ in previous matrix.
Is that all there is to it or are there more things to discover?
The layer of $S^{-1}$ really amounts to the observation that similarity preserves addition and multiplication (and consequently, inverses). That is, the map $X \mapsto SXS^{-1}$ is a homomorphism of algebras. Since this map is invertible, it is also an isomorphism.
Stripping that away, you have observed that multiplying a "type-A" matrix with a "type-B" matrix produces another "type-B" matrix. I'm not really sure what to make of that, honestly. Maybe there's something to say in the case where $C,D$ belong to a class of matrices that is closed under multiplication. For instance, where both $C,D$ are orthogonal/unitary.