There are so many spectrum objects in mathematics.
- spectrum of a linear operator
- spectrum of a ring
- spectrum of a graph
- spectrum in algebraic topology
- spectrum of a sentence (mathematical logic)
There has to be some universal property uniting all of these spectrum objects. How do you use a universal construction to derive each of the five?
This does not answer the question (I'd be surprised if there was a categorical way to define a "spectrum" which somehow encompasses the weird definitions in logic, topology, etc.) but it's a bit too long to be a comment.
It's a bit misleading to think of the first three constructions as the same universal construction going on in three different categories, because they are in fact the same construction in one category: the category of rings. (I wouldn't be surprised if the spectrum in the sense of topology is also the special case of the spectrum of a ring, but I don't know anything about that.)
Let $k$ be an algebraically closed field (characteristic $0$ if you like), and $T$ a linear operator acting on a finite-dimensional $k$-vector space. Then $\operatorname{Spec} T$ is, by definition, the space of "generalized eigenvalues" (i.e. diagonal entries in the Jordan canonical form of $T$) with repetition. But if $\chi_T(x) \in k[x]$ is the characteristic polynomial of $T$ then the roots of $\chi_T(x)$ are precisely the generalized eigenvalues with repetition, so the affine scheme $\operatorname{Spec} k[x]/(\chi_T(x)) = \operatorname{Spec} T$ in a natural way: namely, its points are generalized eigenvalues of $T$, and we can recover their multiplicities as the dimension of the localization as a $k$-algebra. Therefore the spectrum of a ring is a generalization of a spectrum of a linear operator.
Since the "spectrum" of a graph is the spectrum of its adjacency matrix $M$, this is obviously a special case of the spectrum of a linear operator. So it is the spectrum of the ring $\mathbb C[x]/(\chi_M(x))$.