Example 1 Find all $n$ such that there exist $a$ and $b$ satisfying
$$
S(a)=S(b)=S(a+b)=n .
$$
From property (1), we know
$$
a \equiv b \equiv a+b \equiv n(\bmod 9) \text {. }
$$

Thus, $2 n=n(\bmod 9)$.
Therefore, $n$ is a multiple of 9.
When $n=9 k$, take $a=b=10^{k}-1$, at this time,
$$
S\left(10^{k}-1\right)=S\left(2\left(10^{k}-1\right)\right)=9 k \text {, }
$$

satisfying the problem's requirements.