It's really a question about convention. Does such a metric have to be linear on each fiber?
Does it have to preserve the natural Euclidean metric up to a constant factor in each fiber (which is a vector space)?
not sure what you mean but each fiber is a vector space with a metric defined on it. Different fibers have there own separate metric and there is generally no way to compare them among different fibers.
There is generally no natural Euclidean metric on a fiber.
If you have a submanifold of another manifold then its tangent and normal bundles inherit a metric from the ambient manifold.
Oops... You're right. But still, does it have to be a constant 2-tensor on each fiber?