Question [Edited]: [See below.] Are the isomorphisms in $(1)$ and $(2)$ (additive) group homomorphisms?
If I'm right, $\text{End}_R(M)$ is a ring, but $\text{Hom}_R\left(M_i^{n_i},M_j^{n_j}\right)$ is only a group (we can't compose two functions in it). However when coming back to the direct sum of $\text{End}_R\left(M_i^{n_i}\right)$ we get a ring again.
So, are we showing $\text{End}_R(M)\cong\displaystyle\bigoplus_{i=1}^k\text{End}_R\left(M_i^{n_i}\right)$ as rings by getting intermediate group isomorphisms?…
Also, where is the hypothesis "finite type" used in the proof? [Edit: This has been answered in the comments.]
Artin-Wedderburn Theorem: Let $M$ be a semisimple $R$-module of finite type. Then $$\text{End}_R(M)\cong\bigoplus_{i=1}^k M_{n_i\times n_i}(D_i)$$ for some division rings $D_i$.
Proof: $M=\displaystyle\bigoplus_{i=1}^k (M_i)^{n_i}$ where $M_i$ is simple.
\begin{align}
\text{End}_R(M)&=\text{End}_R\left(\bigoplus_{i=1}^k(M_i)^{n_i}\right)\\
&\cong\bigoplus_{1\leq i,j\leq k}\text{Hom}_R\left(M_i^{n_i},M_j^{n_j}\right)\tag{1}\\
&\cong\bigoplus_{i=1}^k\text{End}_R\left(M_i^{n_i}\right)\tag{2}\\
&\cong\bigoplus_{i=1}^kM_{n_i\times n_i}\left(\text{End}_R(M_i)\right)\tag{3}\\
\end{align}
where $(2)$ is a consequence of Schur's Lemma and $(3)$ follows from a previous result.
$M_i$ simple $\stackrel{\text{Schur}}{\implies}$ $\text{End}_R(M_i)$ is a division ring.
All in all,
$$
\text{End}_R(M)\cong\bigoplus_{i=1}^kM_{n_i\times n_i}(D_i)
$$
for some division rings $D_i$.$\blacksquare$
Best Answer
It sounds a bit like by "are they group isomorphisms" you are really asking "are they only group isomorphisms and not ring isomorphisms?" Of course they are additive group isomorphisms, but they are actually more than that.
While the individual Hom groups aren't rings, the sum $\bigoplus_{1\leq i,j\leq k}\text{Hom}_R\left(M_i^{n_i},M_j^{n_j}\right)$ is a ring!
Perhaps the way to get comfortable with it is to imagine it as a "formal" ring of matrices with entries from the Hom groups. Then formal matrix multiplication composes them in a reasonable way such it is a ring. The matrix multiplication matches the composition in $\mathrm{End}_R(M)$ exactly through the map, so it is a ring homomorphism. This holds even if the $M_i$ lack their special properties (being pairwise nonisomorphic simple modules.)
In the passage from line (1) to (2), we simply eliminate a lot of superfluous terms in the direct sum and discover that the multiplication boils down to coordinatewise multiplication of a direct product of rings. That is what the pairwise nonisomorphic condition buys us.
So, the isomorphisms involved are ring isomorphism all the way through.
Here's another sort of example. Let $e$ be an idempotent in any ring with identity, and let $f=1-e$. Then one can show that $R\cong \begin{bmatrix}eRe&eRf\\fRe&fRf\end{bmatrix}$ as rings, where the multiplication on the right is formal matrix multiplication. Neither $eRf$ nor $fRe$ has to be a ring, but $eRe$ and $fRf$ are both rings.
You can look upon this as $R\cong\mathrm{End}_R(R)\cong \mathrm{End}_R(eR\oplus fR)=\mathrm{Hom}_R(eR\oplus fR,eR\oplus fR)\cong\begin{bmatrix}\mathrm{Hom}_R(eR,eR)&\mathrm{Hom}_R(fR,eR)\\\mathrm{Hom}_R(eR,fR)&\mathrm{Hom}_R(fR,fR)\end{bmatrix}$
The (group) isomorphism $\mathrm{Hom}_R(fR,eR)\cong eRf$ can be found in Lam's First course in noncommutative rings (Proposition 21.6.) It turns out to be a ring isomorphism if $e=f$.