It is easy to see the functor $\mathrm{Hom}(A,-)$ commutes with every arbitrary direct sum (i.e. $\mathrm{Hom} (A,\oplus_{i\in I} N_i)=\oplus_{i\in I}\mathrm{Hom}(A,N_i)$) for finitely generated module $A$ since the image of $A$ is contained in finitely many $N_i$. The converse is false, as illustrated in this answer. But I think if we impose projectivity the converse is indeed true. I would appreciate if someone could give me a hint.
Edit: Thanks for the reference provided in the comment (can be found here). However, after I checked it I can't find any assertions related to projective modules. The closest I can found is that if the base ring is Noetherian, then every module satisfies this property will be finitely generated. I am sorry if I missed something since I am not familiar with French.
I think I saw it now. The key fact is every projective module $A$ is a direct sum $\bigoplus_{i\in I} A_i$ of countably generated modules. So I just iteratively pick one module from each summand $A_i$ and form their direct sums. This give an increasing sequence of proper submodules $M_1\subset M_2\subset ...$ whose union is the whole module. Then the quotient map $A\to \bigoplus A/M_k$ cannot correspond to any map in $\bigoplus \mathrm{Hom}(A, A/M_k)$.