This is how I went about the problem, but I'm wanting a second opinion.
Prove by contradiction. Suppose the identity is not unique. Let $a,b\in S$ such that $a$ and $b$ are identities in $*$. Then $a*b=a$ and $a*b=b$ so
$a=a*b=b \implies a=b$
but $a\neq b$ because the identity is not unique.
Therefore, the identity must be unique.