The requirement of a local product structure rests on the bundle having a topology.
Without this requirement, more general objects can be considered bundles.
For instance, the empty function defines a bundle.
If the category is not concrete, then the notion of a preimage of the map is not necessarily available.
Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do; for instance, for a category with pullbacks and a terminal object 1 the points of B can be identified with morphisms p:1→B and the fiber of p is obtained as the pullback of p and π.