Why is HCl not considered to have hydrogen bonding? A molecule that has hydrogen bonding usually follows these two premises.

1.) There is a hydrogen atom involved
2.) Hydrogen must be bonded to a highly electronegative element which are nitrogen ($\ce{N}$), fluorine ($\ce{F}$) and oxygen ($\ce{O}$).

Seeing that both oxygen and chlorine have a small difference in their electronegativity (oxygen being roughly 3.5 and chlorine being roughly 3.0), why does chlorine in a hydrogen chloride molecule ($\ce{HCl}$) have a dipole-dipole interaction, while the oxygen in a water molecule ($\ce{H2O}$) causes the water molecule to have a stronger form of dipole-dipole interaction called hydrogen bonding? I do not understand this since chlorine is just as electronegative as oxygen and nitrogen?
Edit: I should also added that nitrogen and chlorine have the same EN value (3.0)
Very, very related. Check it for a more in depth answer.

Well, first-of-all you need to understand that the $\ce{H}$ bonding isn't actually bonding. It is just a covalent attraction. Also since $\ce{Cl}$ is larger than $\ce{N}$, $\ce{F}$ and $\ce{O}$ it does not make a strong $\ce{H}$ bond. The size of the $\ce{Cl}$ makes the dipole-dipole attraction weaker. However $\ce{N}$, $\ce{F}$ and $\ce{O}$ are smaller and thus have an $\ce{H}$ bond.

Although in reality, compared to other covalently bonded structures, $\ce{HCl}$ has a very strong covalent bond.