If we have the polarization of the electric field in the $z$ direction then we have the selection rule that: $$\Delta m=0$$ (source: slide 7 lecture 19 on this page http://butane.chem.illinois.edu/sohirata/)
What is the physical reason behind this rule?
Since if the emitted photon travels along the $z$-axis we must have $\Delta m =\pm 1$ does this mean that we a photon cannot be emitted in the direction of the polarization of the electric field?
A linearly polarized photon is an eigenstate of angular momentum $m=0$ along the polarization direction. That succinctly explains the selection rule. In your question, you make the error of assuming that the propagation direction and the polarization are both along $z$. In fact, this is impossible, since electromagnetic waves are transverse. If the the electric field direction is along $z$, then the propagation direction lies in the $xy$-plane. For concreteness, let us say the propagation is along $x$. Then the linearly polarized wave is an equal superposition of $m_{x}=1$ and $m_{x}=-1$ circular polarization states; the superposition of these angular momentum states is exactly an $m_{z}=0$ state.
