Why is the zero polynomial not assigned a degree?

This is to make nice rules such as $$ \text{deg }(PQ) = \text{deg }P + \text{deg }Q\\ \text{deg }(P+Q) \le \max(\text{deg }P , \text{deg }Q) $$

So the only value that makes it possible is $$\text{deg }0= -\infty$$

Assigning a degree to the zero polynomial will cause trouble with important and useful theorems that relate the degree of a polynomial to its roots:

If $F$ is a field (examples of fields are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}$, $\mathbb{Z}/p\mathbb{Z}$), a polynomial $P$ with coefficients in $F$ (the set/ring of these polynomials is usually denoted by $F[x]$) of degree $n$, has at most $n$ distinct points $\alpha\in F$ such that $P(\alpha)=0$.

This theorem follows from the fact that we can repeatedly factor out terms of the form $x-\alpha$ (where $\alpha$ is a root) from $P(x)$, lowering the degree of the remaining polynomial by $1$ in each step. See also: http://en.wikipedia.org/wiki/Factor_theorem.

When we restrict to polynomials with coefficients in $\mathbb{C}$, the statement is related to the Fundamental Theorem of Algebra.

Now since the zero polynomial in $F[x]$ has a root at every point in $F$, at least for infinite fields ($\mathbb{R},\mathbb{C},\mathbb{Q}$), we cannot assign a finite value to the degree of the zero polynomial without getting into trouble.