Let $(E,\nabla^E),(F,\nabla^F)$ be smooth vector bundles over a manifold $M$. We say a bundle isomorphism $\Phi:E \to F$ preserves metricity if it "sends" $\nabla^E$-compatible metrics to $\nabla^F$-compatible metrics.
(By "sends" I mean "induces in the natural way" via pullback).
We say $\Phi$ respects the connections if $\nabla_x^F (\Phi(s))=\Phi( \nabla_x^E s)$ for every $s \in \Gamma(E),X \in \Gamma(TM)$
(or equivalently $\Phi$ is $\nabla^{E^* \otimes F}$-parallel: $\nabla^{E^* \otimes F}\Phi=0$, where $\nabla^{E^* \otimes F}$ is the tensor product connection of $\nabla^E,\nabla^F$).
It's clear that every bundle isomorphism which respects the connections, preserves metricity.
(Since then, for any $\nabla^E$-compatible metric $\eta$, $\Phi:(E,\nabla^E,\eta) \to (F,\nabla^F,\phi_*\eta)$ preserves all the structure).
Question: Are there always bundle isomorphisms which preserve metricity, but do not respect the connections?
(There are trivial cases where this holds, e.g when $\nabla^E$ has no compatible metrics. This seems to be interesting when there are "many" $\nabla^E$-compatible metrics).
An equivalent formulation:
A short calculation shows $\Phi$ preserves metricity if and only if
$$ \langle (\nabla_x \Phi)(s),\Phi(t) \rangle_{\Phi_*\eta} + \langle \Phi(s),(\nabla_x \Phi)(t) \rangle_{\Phi_*\eta} =0$$ for every $\nabla^E$-compatible metric $\eta$.
($s,t$ are sections of $E$).
Equivalently: $ \langle (\nabla_x \Phi)(t),\Phi(t) \rangle_{\Phi_*\eta} =0 \iff \langle \Phi^{-1}(\nabla_x \Phi)(t),t \rangle_{\eta} =0$
Denoting $\Psi=\Phi^{-1}(\nabla_x \Phi):(E,\eta) \to (E,\eta)$, we conclude $\Phi$ preserves metricity if and only if
$$\langle \Psi(t),t \rangle_{\eta} =0 $$ which is equivalent to $\Psi^T=-\Psi$ (where the transpose is taken w.r.t $\eta$).
Thus, $\Phi$ preserves metricity iff $\Phi^{-1}(\nabla_x \Phi):E \to E$ is skew-symmetric w.r.t every $\nabla^E$-compatible metric $\eta$. (Note this metrics form a cone over $\mathbb{R}$).
So, the question amounts to the following: Are there always bundle isomorphisms satisfying the above condition, but not parallel?