Hydrogen bonding in chloral hydrate
As noted in the comments, the fallacy is the notion that only a select few atoms can form hydrogen bonds with protic hydrogen.  In reality, the strongest hydrogen bonds involve nitrogen, oxygen, or fluorine, but most nonmetals can form such bonds if they have an electron pair to donate to the bond.  Hydrogen bonding with chlorine is seen in the $\ce{HCl_2^-}$, which is analogous to the more familiar bifluoride ion, as well as chlorine-bearing organic compounds.