If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$? Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define 
$$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E\cong F,$ I mean that $E$ and $F$ are isometrically isomorphic.

Question: Suppose that $p,q\in [1,\infty).$ If 
  $$E\oplus_p E \cong E\oplus_q E\,$$
  then is it true that $p=q$?

If $E$ is of finite-dimensional, then the question is affirmative. 
However, I do not know what will happen if $E$ is of infinite-dimensional.
I would be glad to see a proof if it is true or a counterexample if it is false. 

We say that a norm $\phi:\mathbb{R}^2\to\mathbb{R}$ is normalized if 
$$\phi(0,1) = \phi(1,0) = 1.$$
Also, $\phi$ is monotone if for $|a_1|\leq |b_1|$ and $|a_2|\leq |b_2|,$ then 
$$\phi(a_1,a_2) \leq \phi(b_1,b_2).$$
We define 
$$E\oplus_\phi E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \phi(\|x\|, \|y\|) \}.$$
A more general question: 

Suppose that $\phi,\psi:\mathbb{R}^2\to \mathbb{R}$ are norms that satisfy normalization and monotonicity.
  Assume that $\phi$ and $\psi$ are not $\ell^\infty$ norm.
  If 
  $$E\oplus_\phi E \cong E\oplus_\psi E,$$
  then is it true that $\phi = \psi?$?

 A: 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.
A: It was proved by E. Behrends in Studia Math. 55, 71-85 (1976) that apart from $E=\mathbb{R}^2$ with the sup norm, which is isometric to $E$ with the $\ell_1$-norm, a Banach space $E$ admits a decomposition $E_1\oplus_p E_2$ for at most one value of $p$. This theorem is what Misha has outlined above. 
