If you are measure the running time in $(m,n)$ then $O(mn)$ is not linear function in $(m,n)$. If there is no relation between $m$ and $n$ this function can grow quadraticly in general.

However it is a linear function in each of them separately, i.e. if you fix one of them and look at the growth in the other variable then it is a linear function in other one.

If you are measuring the running time in $(m,n)$ then $O(mn)$ is not a linear function in $(m,n)$. If there is no relation between $m$ and $n$ this function can grow quadraticly in general.

However it is a linear function in each of them separately, i.e. if you fix one of them and look at the growth in the other variable then it is a linear function in other one.