As Zhen Lin says, a really important part of the Eilenberg–Zilber theorem is that the lax-monoidal structure map is a quasi-isomorphism. Nevertheless, reformulating the first part of the theorem as "the functor $C_*(-)$ is lax-monoidal" is helpful for applying general results to it. For example:
Proposition. Let $\mathsf{C}$ and $\mathsf{D}$ be symmetric monoidal categories, and $F : \mathsf{C} \to \mathsf{D}$ a lax monoidal functor. Then $F$ induces a functor from monoid objects in $\mathsf{C}$ to monoid objects in $\mathsf{D}$.
Proof. Let $M$ be a monoid object in $\mathsf{C}$. The product morphism on $F(M)$ is given by
$$F(M) \otimes F(M) \xrightarrow{\text{lax monoidal}} F(M \otimes M) \xrightarrow{F(\mu)} F(M)$$
and the unit by
$$1_\mathsf{D} \xrightarrow{\text{lax monoidal}} F(\mathsf{1}_C) \xrightarrow{F(e)} F(M).$$
Then it's easy to check that this gives $F(M)$ the structure of a group object, and that $F$ then defines a functor from group objects to group objects.
Corollary. If $M$ is a topological monoid, then $C_*(M)$ is a monoid in the category of chain complexes, i.e. a dg-algebra.
The proof is immediate from the Eilenberg–Zilber theorem, and it's helpful to split the proof in two parts in order not to get bogged down in technical details. More generally, if $\mathtt{P}$ is a topological operad, then $C_*(\mathtt{P})$ is a dg-operad, and if $A$ is a $\mathtt{P}$-algebra, then $C_*(A)$ is a $C_*(\mathtt{P})$-algebra.
Basically, any sort of "product" structure can be transported from topological spaces to chain complexes; in general, this is "the point" of determining if a functor is lax monoidal or not. You would need an oplax monoidal functor to transform topological "coproduct" structures into coproduct dg-structures. The EZ functor isn't oplax monoidal, so in general you cannot expect it to turn "coalgebras" into "coalgebras".
