Why are acids characterized by giving hydrogen and bases characterized by receiving it when acid and base reactions are double displacement reactions? One example of this is $\ce{HCl}$ and $\ce{KOH}$. Why would the acid not just give the hydrogen ion in $\ce{HCl}$ to form $\ce{Cl-}$ and $\ce{KOH2+}$?
$\ce{KOH}$ does not exist as a molecule.  In water it is completely dissociated as $\ce{K+}$ and $\ce{OH-}$.  It does not make sense to write $\ce{KOH_2^+}$ for this reason.