It is a sketch of the proof in the case of $p\ne q$ (a complete proof on these lines is rather lengthy). Something similar can be done in more general case, possibly with some exceptions.
Our plan is the following: We assume that $p,q\in[1,\infty)$, $p\ne q$ and $W=E_1\oplus_p E_2=F_1\oplus_q F_2$ (isometrically), where $E_1,E_2,F_1,F_2$ are all isometric to a Banach space $E$, and get a contradiction.
By $S(X)$ we denote the unit sphere of a Banach space $X$, if $Z$ and $Y$ are subspaces of $X$, we set $\delta(Y,Z)=\inf\{||y-z||:~ y\in S(Y), z\in Z\}$ and call it an inclination of $Y$ to $Z$.
If we can find $i\ne j\in\{1,2\}$ such that $F_i$ has zero inclination to $E_1$ and $F_j$ has zero inclination to $E_2$, we get a contradiction by observing that it means that two-dimensional $\ell_p$-sphere approximates $\ell_q$-sphere with an arbitrary precision, which is not true ($p$ and $q$ are fixed).
[This paragraph is added on 8/8/18 according to the request below] Assume without loss of generality that $\delta(F_1,E_1)=\delta(F_2,E_2)=0$. This implies that there are sequences $\{x_i\}\in S(F_1)$, $\{y_i\}\in S(E_1)$, $\{z_i\}\in S(F_2)$, and $\{w_i\}\in S(E_2)$, such that $$\lim_{i\to\infty}||x_i-y_i||=0=\lim_{i\to\infty}||z_i-w_i||=0$$ Observe that our assumptions imply that the subspace $A_i$ spanned by $\{x_i,z_i\}$ is isometric to $\ell_q^2$ and the subspace $B_i$ spanned by $\{y_i,w_i\}$ is isometric to $\ell_p^2$. Therefore, using the argument of Proposition 1.a.9 in Lindenstrauss-Tzafriri, Classical Banach spaces, v. I we see that there should be isomorphisms $T_i$ between $\ell_p^2$ and $\ell_q^2$ mapping unit vector basis of $\ell_p^2$ onto the unit vector basis of $\ell_q^2$ and being arbitrarily close to the identity. This is clearly false if $p\ne q$.
It remains to consider the case where both $F_1$ and $F_2$ have nonzero inclination to $E_1$ (or to $E_2$, the cases are similar).
By $P_1$ and $P_2$ we denote projections on $W$ corresponding to the decomposition $E_1\oplus_p E_2$. Nonzero inclination to $E_1$ implies that the restriction of $P_2$ to both $F_1$ and $F_2$ are isomorphic embeddings. If there are nonzero points $y_1$ and $y_2$ in $F_1$ and $F_2$ which have the same image in $E_2,$ we get a contradiction by considering the space spanned by $y_1$ and $y_2$: on one hand its unit sphere is $\ell_p$-sphere, and on the other hand - $\ell_q$-sphere.
In a similar way (but using approximations) we get a contradiction in the case where the inclination of $P_2(F_1)$ to $P_2(F_2)$ is zero. Finally, if the inclination of $P_2(F_1)$ to $P_2(F_2)$ is nonzero, we get a contradiction because in this case $P_2$ would be an isomorphic embedding of $W$ into $E_2$, which is obviously false.