Extension Theorems If we were to abstractly represent "analytic continuation" from complex analysis, it would look something like this:
"If $f$ is a function with property $P$ on a set $U$ satisfying some conditions $C$, then $f$ can be uniquely extended to a bigger set $Q$ such that $P\subset Q$ and $f$ still has property $P$."
In measure theory, the Caratheodory Extension Theorem says that any measure $\mu$ defined on a ring $R$ of subsets can be extended to the sigma algebra generated by $R$ and this extension is unique as long as the $\mu$ is sigma finite. 
So in a way, subaddativity and sigma-finiteness for measures are analogous to complex-analytic for functions. 
Question 1: What are some other examples of such unique extension theorems, say in combinatorics, or group theory?
Question 2: Is there a nice unification ("categorification?") of these extension theorems? Specially, in what ways must property $P$ be special?
Edit: I would like to re-emphasize that the extension should be unique. 
 A: Some examples:


*

*Free objects.  There are free groups, abelian groups, rings, modules, algebras, commutative algebras, monads, etc. etc. It is far more than sufficient to have nice left-adjoints in your category. In fact, a vector space is just a module over a field, which is automatically a free module, so both of @szw1710's answer is just a restatement of the universal property for free modules

*Continuous extension to boundary (this is a basic version for real analysis; I can't find the more general version online at the moment, but it can be found in Munkres). Elementary topology shows an extension to a boundary must be unique under relatively weak hypotheses, and slightly stronger hypotheses guarantee this.

*Tietze Extension Theorem. Allows us to extend continuous functions on a closed subset of a normal space to the whole space

*Schwarz Reflection Principle. Allows us to extend harmonic functions by reflecting over some geometric object. The linked version is the simplest case in complex analysis for extending an analytic function on the upper half plane to the whole space, but we can easily extend over any generalized line (e.g. circles).

*Extension to unique, maximal atlas. In (real) differential geometry we may extend our local maps to our underlying space into a map over the whole atlas. The situation is much more complex (no pun intended) over complex manifolds because of the strictness of the holomorphicity condition, but the same principle roughly carries over


Hundreds of these examples spread throughout higher mathematics. Many more examples can be found on this Wikipedia page, You will stumble across many of them as you continue your studies. Much of modern mathematics is us abstracting well studied ideas with the hopes to explain more general phenomena, and extension lemmas are no counterexample; throughout nearly every field of mathematics we wish to define things locally and then extend globally.
You could unify the language for these categorically, but you will essentially just be encoding the properties by enriching your category.
A: A simple example of the unique extension comes from linear algebra. Let $V,W$ be vector spaces over the same field and $B\subset V$ be a basis. Then any mapping $f:B\to W$ is uniquely extendable to a linear map $F:V\to W$.
In group theory look at free groups.
