The point of view of making all choices at once instead of canonical choices is part of Grothendieck's philosophy. In SGA1, VI, 12, after considering the notions of fibered categories and cleavages, we can see the following remark at the end:
"[...] Il est d’ailleurs probable que, contrairement à l’usage encore prépondérant maintenant, lié à d’anciennes habitudes de pensée, il finira par s’avérer plus commode dans les problèmes universels, de ne pas mettre l’accent sur une solution supposée choisie une fois pour toutes, mais de mettre toutes les solutions sur un pied d’égalité."
which essentially qualifies the "canonical choice" approach as an ancient old form of thinking, stressing the preference for the "making all choices at once" point of view. This is particularly well illustrated in the case of the concept of cleavages of Grothendieck fibrations. One of the first examples of fibered categories (probably the one upon which Grothendieck was thinking when coming up with this concept) is the one associated to the slice categories with pullback functors between them. Since generally there's no canonical choice of pullbacks, the existence of pullback functors usually appeals to the axiom of choice. This amounts to specifying a cleavage in the fibration considered, but Grothendieck wanted to avoid, when possible, working with cleavages.
A case where the above is relevant is the following. In 1978 Joyal gave a series of lectures in Montréal exposing a categorical proof of the completeness theorem for many kinds of logic (including classical logic). It was essential for Joyal's proof, in the case of non classical logics, not to rely on the axiom of choice, and hence Grothendieck fibrations without cleavages was the solution for making his proof constructive. This has though the disadvantage of making the proof less intuitive, and in general related expositions (as can be seen for example at Johnstone's "Sketches of an Elephant", D1.5) prefer to use the "canonical choice" approach that is a bit less obscure.