Let $\text{diag}(\mathbf{M})$ denote the column vector containing the diagonal elements of $\mathbf{M}$. Then, from \begin{equation} \text{diag}(\mathbf{A}\mathbf{B}) = \text{diag}(\mathbf{A}\mathbf{C}) \,, \end{equation} what can be said about $\mathbf{B}$ and $\mathbf{C}$?
Is $\mathbf{C} = \mathbf{B}$ the only solution? Or are there many choices of $\mathbf{C}$ for any particular $\mathbf{B}$? What do these choices look like?
Edit: I am interested in the case when $\mathbf{A}$ is full-rank.
We cannot say much about it : if $A$ is the zero matrix, then any matrices $B$ and $C$ will verify $diag(AB)=diag(AC)$.
Also, even if $A$ is invertible, say $A=I$ the identity for example, then any pair of matrices $B$ and $C$ with same diagonal will verify $diag(AB)=diag(AC)$ but $B$ and $C$ can be very different.
More generally, when $A$ is full rank as you wanted in your edit, take $B=A^{-1}B'$ and $C=A^{-1}C'$ where $B'$ and $C'$ are matrices with same diagonal but with different entries elsewhere, then $diag(AB)=diag(AC)$.