The bichloride ion $\ce{HCl2^-}$ does exist, but it requires other environment than usual solutions in water. E.g. solution of $\ce{[N(CH3)4]Cl}$ salt and undissociated $\ce{HCl(solv)}$ in nitrobenzene solvent, as mentioned in the ACS link of the Oscar's answer. It is the scenario that makes the partial hydrogen charge in $\ce{H^{(+)}-Cl{(-)}}$ quite favourable for chloride anions.
For water solutions scenario, the key is the presence of undissociated $\ce{HF}$ and the ability of $\ce{F-}$ to make hydrogen bond with $\ce{HF}$. Neither is available for $\ce{HCl}$ and $\ce{Cl-}$. There are just very minor traces of undissociated $\ce{HCl}$ in water and $\ce{Cl-}$ would not have bound to it even if it had been there.
There are no $\ce{HCl}$ or $\ce{KCl}$ in water as molecules. There are hydrated ions $\ce{H+(aq)},$ $\ce{K+(aq)}$ and $\ce{Cl-(aq)}.$ Hydrated molecules $\ce{HF(aq)}$ are there in hydrofluoric acid or in acidified fluoride solutions, reacting:
$$\ce{F-(aq) + HF(aq) <=> HF2^-(aq)}\tag{R1}$$
Hydrofluoric acid is weak mainly because of forming the stable ionic pair, so ion $\ce{H3O+(aq)}$ is mostly not free:
$$\ce{HF(aq) + H2O(l) <=> H3\overset{+}{O}\bond{...}F-(aq) <=> H3O+(aq) + F-(aq)}\tag{R2}$$
The structure of $\ce{HF2-}$ is $\ce{[F\bond{...}H-F]-}.$ There is also $\ce{[H-F\bond{...}H]+}$ in concentrated hydrofluoric acid or liquid hydrogen fluoride:
$$\ce{3 HF <=> H2F+ + HF2^-}\tag{R3}$$
Liquid hydrogen fluoride has such a high boiling point due $\ce{HF}$ chain linked by hydrogen bonds:
$$\ce{[\bond{...}H-F\bond{...}H-F\bond{...}H-F\bond{...}H-F\bond{...}]}$$