Sorry if this isn't the right place for this, it hasn't gotten any answers on ME. I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are determined up to unique isomorphism, algebraic closures (and by extension, their Galois groups) are determined only up to automorphism (conjugation). It seems to me that there ought to be some interpretation of this in terms of bicategories (weak 2categories). This intuition is supported by the fact that 2cells are given by conjugation when we give Grp the structure of a 2category (view groups as 1object categories, get 2cells via natural transformations). Is there any such interpretation?

I don't see bicategories coming into this in a useful way, but I think what you have is a consequence of two more general facts:
With the objects satisfying a universal property the comparison isomorphisms between them are unique, so that the groupoid of objects satisfying the universal property is codiscrete, i.e. there is exactly one morphism between any two objects, so in particular the vertex groups of this groupoid are trivial. For an object A in a concrete category (or in a category with a chosen class of 'embeddings') there is a groupoid of injective hulls of A that is not in general codiscrete, and so it can have nontrivial vertex groups. But this groupoid, though not codiscrete, is still connected, so that each vertex group is (nonuniquely!) isomorphic to every other via conjugation by a morphism (necessarily invertible) of injective hulls. Edit: The fundamental group of a space, as in JSE's analogy, bears much the same relationship with the fundamental groupoid of the space  in particular, $\pi_1$s at points in the same pathcomponent are isomorphic via conjugation in the same way. 


Not necessarily. In the higher categorytheoretic setting one asks for a 'contractible space' of choices (space might mean simplicial set or ncategory) instead of uniqueness. The natural 2category one might define may not be the 'right' one to get such a collection of choices, and so one could define a 2category such that these things are unique in the appropriate sense, but this might just be cooked up to give that result and not of intrinsic interest. For example, one can define the 2category of fields where the underlying 1category is $Fields$, and there is a unique 2arrow between any two parallel 1arrows. This is clearly not an interesting 2category. (Other examples of nonunique closures are given here: http://nlab.mathforge.org/nlab/show/completion#nonunique) 

