For $V$ finite-dimensional, the algebra $\hom(V, V) \cong V^\ast \otimes V$ is naturally self-dual, and this duality may be used to transfer the algebra structure on $\hom(V, V)$ to a coalgebra structure on $\hom(V, V)$, and vice-versa. Notice that the duality functor $\text{Vect}^{op} \to \text{Vect}$ induces a functor from coalgebras to algebras.
Thus a coalgebra map $M_n(\mathbb{C}) \to M_{n+1}(\mathbb{C})$ would induce a dual algebra map $M_{n+1}(\mathbb{C}) \to M_n(\mathbb{C})$. But even when $n=1$ there is no such algebra map.
I don't know whether the space of compact operators carries a (natural) coalgebra structure.