When I try to learn about stacks, something that is mentioned all over the place (as a reason not to define moduli problems as functors from schemes to groupoids) is the fact that the pullback of a family along a morphism of schemes is not well-defined, but is only defined up to unique isomorphism.
1) Is the fibered product of two maps of schemes well-defined? The tensor product of two rings is certainly well-defined, but maybe this means the fibered product is only well-defined after we make a choice of affine open sets? Or is it that the choice doesn't work with composition because $(A\otimes_RB)\otimes_RC$ is only canonically isomorphic to $A\otimes_R(B\otimes_RC)$?
2) If the answer to the previous question is yes, what is an example of a "geometric" moduli problem where there is no obvious choice of pullback of families? I'm familiar with an example from http://people.brandeis.edu/~tbl/minor-thesis.pdf, in which a choice of pullback corresponds to a splitting of some map of groups. But I can't see a way to construct an example like this where the objects considered look like "families of geometric objects."
1) There is a global construction of the fiber product of schemes which does not depend on affine covers and actually also works for locally ringed spaces; see my answer here. I think that in all examples in practice there will be specific constructions of universal objects (such as fiber products) and there is no problem to define them as functors. If one does not want to fix a version but would like to look at all of them, one might use anafunctors. In any case, it doesn't really matter at all since universal objects are unique up to unique isomorphism (subject to compatibility conditions). One should always remember that universal objects may have many constructions (see here for a construction of the tensor product of modules which doesn't use free modules), but in the end it doesn't really matter which one we choose. In fact we may just forget the whole construction and use only the universal property.
2) I am not an expert, but I doubt that there will be any geometric examples. By the way, in my research I have always worked with (algebraic) stacks as $2$-functors $\mathsf{Sch}^{op} \to \mathsf{Gpd}$ and although this might be naive, it worked out fine so far ;).