Why there is no bichloride or binitrate ion like bicarbonate or bisulphate?
Okay, you want simple, here's a real simple way to put it.

How many limbs does a dog have? 4 right?
Now if it loses one of its limbs, a prosthetic one can be placed,(We'll get to this: with its whole body, being able to cope with the leg). Now consider the case of a human being, same scenario, the same thing can be done, since we can place the limb in the "slot"(somebody put a better word here), where the leg previously was(although, the limb cant do everything a leg couldve done). Now consider the case of a creature with just one limb, if it loses the sole limb it had, life's gonna be hard, simply because of the amount of discomfort its going to face while adapting to the limb. Similarly, sulphate ($\ce{SO4^2-}$) and carbonate ($\ce{CO3^2-}$) have a valency greater than 1. Hence can accept an H+ ion and still exist as a stable specie, and further accept a metal ion;

In the case of $\ce{HCl}$, it has a valency of just 1, hence upon accepting an $\ce{H+}$ to get the "bi" tag to its name, it has lost its charge, and now is incapable of accepting a metal ion to form a salt(while keeping the $\ce{H+}$ to itself), as it has simply become hydrochloric acid.