Let $E$ be a vector bundle over a manifold $M$. By definition, a metric on $E$ is a function $g:E \times _M E \to M \times \mathbb{R}$. In wikipedia, they say $g$ is a bundle map.
However, it is not linear on the fibers. The fiber $E_p \times E_p$ has the vector structure of the direct sum, while $g$ is bi-linear. (Hence $g_p$ is not linear).
So, $g$ is a morphism of fiber bundles, not of vector bundles.
Wouldn't it be more natural to define $g:E \otimes E \to M \times\mathbb{R}$ where now it is linear on the fibers?