Does a covalent bond constitute of 1 or 2 electrons?
The bond in your picture is made up of two electrons, and in the Lewis formalism so is every single bond. Yet there is no negative charge. We can think of this in two ways:

Remember that the bond electrons are shared. So each atom contributed one electron and, upon dissociation, would usually take back that one electron. Counting only that one electron will give a neutral hydrogen atom, not a hydride ion.

Charges on an atomic or molecular level are due to a mismatch of the numbers of protons and electrons. In the hydrogen molecule, you have two nuclei consisting of one proton each, meaning an overall $2+$ charge. Furthermore, you have two electrons that get shared in the covalent bond giving an overall $2-$ charge. $+2 + -2 = 0$, so the overall molecule is neutral.
If you wanted to place both electrons on one atom to form a hydride anion, that would mean the other atom is left with zero. Instead of a hydrogen molecule $\ce{H-H}$ you would have a separated ion pair $\ce{\overset{-}{H}\bond{...}\overset{+}{H}}$.


Once you leave the very simplified Lewis formalism and turn to theories such as molecular orbital theory which explain molecules much more accurately, this way of thinking will be lost altogether since there is no longer an obvious localisation of two electrons to a single bond.