If $E \to M$ is a tensor bundle over a Riemannian manifold (M, g) then the metric $g$ induces an inner product on the fibers of $E$ in the same way that one can give tensor products of inner product spaces an inner product.
We can also consider direct sums of tensor bundles which will also have an inner product induced from the metric in the same way that the direct sum of some inner product spaces is an inner product space with the inner product taken to be the sum of the inner products on the projections.
In this way one can build a collection of vector bundles over $M$ which have some sort of natural inner product structure induced from the metric $g$. Does this collection of vector bundles have a name and is this talked about in any texts?
I do not know if this is what you are looking for, but this recall me the notion of "bundle metric". This is treated in Jost's Riemannian Geometry and Geometric Analysis, chapter 2. Another book (but very difficult to find unless you live in Brazil) is "Tópicos de Geometria Diferencial" written by Antonio Caminha and Muniz Neto. In this last reference, they call the bundles with metric something like "Riemannian bundles".