Thanks for the question. I can see what you are getting at.
The uncontroversial (I hope!) bit is that it is a special case of dipole-dipole interactions. What makes hydrogen bonds stronger is that the lone pairs on the electronegative atom (N, O, F) are involved in the attraction to the δ+ hydrogen.
We use this definition at A Level but if we think about it logically, why should it not apply to ions? So when we have hydroxide ions and water, what is the interspecies bonding going to be?
The author of Chemguide says: "A hydrogen bond is made from two parts - a δ+ hydrogen attached to a sufficiently electronegative element, and an active lone pair." And then extends that to explain hydrogen bonding with negative ions.
(The sufficiently electronegative element and an active lone pair bit limits it to N, O and F in polar molecules.)
Cl won't form hydrogen bonds normally in a polar molecule as the lone pairs are at the 3-level and are not active enough. As the chloride ion, with a negative charge, the lone pairs are active enough