Call a pair of topological spaces $(X,A)$ a good pair if $A\subseteq X$ is a closed subspace and there exists an open neighborhood $A\subseteq U \subseteq X$ such that $A$ is a deformation retract of $U$. For any good pair $(X,A)$ one has $H_n(X,A)\cong \tilde{H}_n(X/A)$ for all $n\geq 0$.
Using the long exact sequence of a pair, this rather immediately implies $\tilde{H_n}(\bigvee_i X_i)\cong \bigoplus_i \tilde{H_n}(X_i)$ if the pair $(\coprod_i X_i, \coprod_i\{x_i\})$ is good. Here $x_i$ is the chosen base point of $X_i$.
Does this result hold even if the pair $(\coprod_i X_i, \coprod_i\{x_i\})$ is not good? In other words is $n$-th reduced homology a coproduct-preserving functor from the category of pointed topological spaces to the category of abelian groups? Is there a counterexample for finite wedge sums?