The electric field from each plate does not act on itself. It only acts on the other plate, and any other charges. The field $E_1$ acts on $q_2$ and the field $E_2$ acts on $q_1$.

If $\sigma_1=\sigma_2$ then the total electric field acting on some other charge is zero, but the field acting on each plate is non-zero so the plates still repel.

The electric field from each plate does not act on itself. It only acts on the other plate, and any other charges. The field $E_1$ acts on $q_2$ and the field $E_2$ acts on $q_1$.

If $\sigma_1=\sigma_2$ then the total electric field acting on some other charge is zero, but the forces $q_1E_2$ and $q_2E_1$ acting on each plate are non-zero so the plates still repel.