Let $(E_i)_{i\in I}$ be a family of $\mathbf{Z}$-modules and let $(J_k)_{1\leq k\leq n}$ be a finite partition of $I$. Suppose $$f:\prod_{k=1}^{n}\prod_{i\in J_k} E_i\rightarrow F$$ is a $\mathbf{Z}$-multilinear mapping. It seems that one can assert that $f$ is really a multilinear map from $\prod_{i\in I}E_i$ using the isomorphism $$\phi:\prod_{i\in I}E_i\rightarrow\prod_{k=1}^{n}\prod_{i\in J_k} E_i,\,x\mapsto(pr_{J_k}(x))_{1\leq k\leq n}.$$
But $f\circ\phi$ is not $\mathbf{Z}$-multilinear..Then, what is the justification for saying that $f$ can be seen as a multilinear map from $\prod_{i\in I}E_i$?
I do not believe you can regard $f$ as a multilinear map from $\prod_i E_i$. Recall that there is a one-to-one correspondence between multilinear maps from a product and linear maps from a tensor product. Thus, multilinear maps out of
$$\prod_{k=1}^n\prod_{i\in J_k}E_i$$
look like linear maps out of
$$\bigotimes_{k=1}^n\prod_{i\in J_k}E_i$$
whereas multilinear maps out of
$$\prod_{i\in I}E_i$$
look like linear maps out of
$$\bigotimes_{i\in I}E_i.$$
However there is no natural correspondence between linear maps out of $\otimes_{i}E_i$ and linear maps out of $\otimes_{k=1}^n\prod_{i\in J_k}E_i$ in general. So I do not believe $f$ can be regarded as a multilinear map out of $\otimes_{i\in I}E_i$. The reason that $\phi$ fails to give a correspondence is that $\phi$ is a linear isomorphism, not a mutlilinear isomorphism.