How to identify the object $K$ in $K-\operatorname{Vect}$ categorically? How does one identify the field $K$ in the category of vector spaces $K-\operatorname{Vect}$ over $K$? The obvious objects are to try are the initial & terminal objects, but these aren't right, as they're both any zero-dimensional space.
The Hahn-Banach theorem shows that $K$ is injective in the category of normed vector spaces over either the field of reals or complex numbers. But is it the only (upto isomorphism) injective object? 
 A: Here is another characterization: It is the "unique" non-zero object which admits a monomorphism into any non-zero object.
It is also the "unique" non-zero  object which is a quotient of any non-zero object.
Actually we can characterize every vector space since we can characterize the dimension function: It is the unique surjective function $\dim : \{\text{vector spaces}\} \to \{\text{cardinals}\}$ such that $\dim(0)=0$ and $\dim(\bigoplus_i V_i) = \sum_i \dim(V_i)$. For example, this means that $K$ is the unique non-zero indecomposable object.
A: Well, for instance $K$ is (up to isomorphism) the only object $X$ in your category such that the monoid $\operatorname{End} X = \operatorname{Hom}(X,X)$ is nontrivial and commutative.    
If we give ourselves the structure of an additive category, $K$ is the unique (up to isomorphism) simple object and also the unique object such that the endomorphism ring $\operatorname{End} X$ is nontrivial and has no zero divisors.  Alternately, it is the unique object such that the endomorphism ring is a division ring.  This latter description loses the commutativity of the previous paragraph, but maybe that is a feature, not a bug: this description would also work if one replaced the field $K$ by any division ring.
Also:

The Hahn-Banach theorem shows that K is injective in the category of normed vector spaces over either the field of reals or complex numbers. But is it the only (up to isomorphism) injective object? 

A zero object is always injective.  Finite direct sums of injective objects are injective.  So no.
Added: Martin Brandenburg correctly points out that the notion of simple object exists in any category.  I think that while it is important to see that $K$ is distinguished in $\operatorname{Vect}_K$ as a completely naked category, thinking in terms of the additive structure is also natural.  As an additive category, $\operatorname{Vect}_K$ is semisimple (every object is a direct sum of simple objects) and has a unique (up to iso...) simple object.  The characterizations given in Martin Brandenburg's answer can both be formulated without using the additive structure...but they are also very naturally viewed as consequences of the previous sentence.  
