【Analysis】The essence of the problem is to find the conditions that $m$ must satisfy for $\mathrm{C}_{m}^{1}$, $\mathrm{C}_{m}^{2}, \cdots, \mathrm{C}_{m}^{m-1}$ to all be multiples of a prime $p$.

Solution Let $p$ be any prime factor of $n$. Since $p \mid \mathrm{C}_{m}^{k}$ if and only if the $p$-adic representation of $k$ has at least one digit greater than the corresponding digit of $m$, $\mathrm{C}_{m}^{1}, \mathrm{C}_{m}^{2}, \cdots, \mathrm{C}_{m}^{m-1}$ are all multiples of $p$ if and only if the leading digit of $m$ is 1 and all other digits are 0, i.e., $m=p^{u}$.
This implies that $n$ has only one prime factor $p$, i.e., $n=p^{v}$.
Taking $k=p^{u-1}$.
Since the $p$-adic subtraction of $m-k$ involves only one borrow, the power of $p$ in $\mathrm{C}_{m}^{k}$ is 1, thus $v=1$.

Therefore, all $(m, n)$ that satisfy the condition are $\left(p^{u}, p\right)$, where $p$ is a prime number and $u$ is a positive integer.
Example 4 Find all pairs of positive integers $(m, n) (n>1)$ such that the binomial coefficients $\mathrm{C}_{m}^{1}, \mathrm{C}_{m}^{2}, \cdots, \mathrm{C}_{m}^{m-1}$ are all multiples of $n$.

保留源文本的换行和格式，翻译结果如下：

Example 4 Find all pairs of positive integers $(m, n) (n>1)$, such that the binomial coefficients $\mathrm{C}_{m}^{1}, \mathrm{C}_{m}^{2}, \cdots, \mathrm{C}_{m}^{m-1}$ are all multiples of $n$.