The following is from The Classical Groups Their Invariants and Representations, by Hermann Weyl:
$f\left[x\right]$ being a polynomial in $x,$ $\alpha$ is a root or zero of $f$ if $f\left[\alpha\right]=0.$ A polynomial of degree $n$ has at most $n$ different zeros; this follows in the well-known way by proving that $f\left[x\right]$ contains the factors $(x-\alpha_{1})(x-\alpha_{2})\dots$ if $\alpha_{1},\alpha_{2},\dots$ are distinct zeros. Hence a polynomial $f\left[x\right]\ne0$ does not vanish numerically for every value of $x$ in $k,$ provided the reference field $k$ is of characteristic $0,$ because such a field contains infinitely many numbers.
That statement seems problematic to me. It appears to say that a polynomial that is non-zero somewhere is not zero everywhere, which is a tautology. Is he really saying that a polynomial whose coefficients are not all zero will not vanish identically on a field of characteristic $0$?
That is either obviously the case, or I am failing to understand what is meant by a polynomial on a field of characteristic $0$.
Yes, that's exactly what he's saying. Note that a polynomial is defined as a sequence of coefficients, not as a function. So $f[x]\neq 0$ means that the sequence of coefficients is not the zero sequence, i.e. that the coefficients are not all zero.