In order to superimpose two or more electric fields, they must be defined in the same region. In other words $$\mathbf E=\mathbf E_1(x_0,y_0)+\mathbf E_2(x_0,y_0)$$ for some point in space $(x_0,y_0)$
In your case, $\mathbf E_1$ and $\mathbf E_2$ exist in different regions of space. Therefore, you can't superimpose them and they don't cancel. It's kind of like asking if I put my car in drive and your put yours in reverse why both of our cars don't just stay still.
Perhaps your confusion lies in thinking that the field vectors in your picture only exist on the sheet. This is not the case. At every point $(x,y,z)$ in space there is an electric field. For $z>0$ the field points upward, for $z<0$ the field points downward. You can't add the upward and downward fields together because they exist in different regions of space.
This is a inside look at the atomic scale:
The field effect is radial and the intersecting field effects sort of cancles out. The direction lines are not forces on the plate but the field effect direction from the plate. You only do that sort of vector cancellation if the force vectors are rather on the plate. If another lone floating positively charged particle falls anywhere in this region, those field lines are the direction of force on the charge. We aren't vectorially resolving the field lines, but rather the force effect on another charged body.