Let us look at two monads on $\bf Set$. The first will be the finite sequence monad (from the free forgetful adjunction with $\bf Mon$.)
$$\eta(x)=[x]$$ $$\mu([[a,b, \dots, z],[\alpha,\beta, \dots, \zeta],\dots,[\circ, \dots, \triangle]])=[a, b, \dots z, \alpha, \beta, \dots, \zeta, \dots, \circ, \dots, \triangle]$$
Notice that, although $\mu$ loses some structure (the specific divisions of the lists), but none of the values are lost.
Now take the infinite sequence monad, with diagonal multiplication.
$$\eta(x)=(x, x, x \dots)$$ $$\mu(((a,b,c, \dots),(\alpha,\beta,\gamma,\dots),(\circ,\square,\star,\dots),\dots)) = (a, \beta, \star, \dots)$$
Notice that know values are being lost ($b, \alpha, \square$, etc...). (Interestingly enough, the structure can be recovered though (although, this is probably because it only has one structure.))
I am wondering if there is a way to formalize this in general. I know in general categories, whether or not a morphism is mono determines if it is losing some information, but $\mu$ always loses some information, namely the structure.
Is there some way to "determine" or generalize if a monad's multiplication is losing any of the underlying values (in the same way mono is a "determine" or generalize if a morphism is injective.)