Assume that for a given set of matrices ${\bf A_i}$ we have some canonical basis
$${\bf A_i = TC_iT}^{-1}$$
Our $\bf C_i$'s are not diagonal, but block-diagonal. For example one block could look like
$$\begin{bmatrix} 1&c_{f(i)}\\0&1 \end{bmatrix}$$
And it will perform addition between these $c_{f(\cdot)}$:
$$\text{Corresponding } {\bf C_i C_j} \text{ block } : \begin{bmatrix} 1&c_{f(i)}+c_{f(j)}\\0&1 \end{bmatrix}$$
The only theory I know about which vaguely resembles this is Jordan blocks in the Jordan Normal Form, but then these blocks have very determined structure, and it is "opposite" as compared to above:
$$\begin{bmatrix} \lambda_{f(i)}&1\\0&\lambda_{f(i)} \end{bmatrix}$$
Does there exist some general theory in linear algebra investigating what happens with arbitrary blocks like the one above?
Let's say that the vector space $V$ has a direct sum decomposition $V = A \oplus B$. Then any endomorphism $f: V \to V$ is expressible as a block matrix with respect to this decomposition: $$ f = \begin{pmatrix} {_A}f_A & {_A}f_B \\ {_B}f_A & {_B}f_B \end{pmatrix},$$ where ${_B}f_A: A \to B$ is the restriction of $f$ to $A$, followed by projection to $B$. If $f$ has the form that you are describing, then $$ f = \begin{pmatrix} {_A}f_A & {_A}f_B \\ {_B}f_A & {_B}f_B \end{pmatrix} = \begin{pmatrix} \operatorname{id}_A & {_A}f_B \\ {_B}0_A & \operatorname{id}_B \end{pmatrix},$$ And so the $f: V \to V$ which come out as block matrices of this form are in bijection with linear maps $g: B \to A$, via the correspondence $g = {_A}f_B$.
So these types of matrices correspond to linear maps $B \to A$, and (as you found), composition of these types of matrices corresponds to addition of linear maps $B \to A$. In fact, I think also inverting this type of matrix corresponds to negating a linear map $B \to A$.