When we say that energy of hydrogen bonds in liquid water is $\approx \pu{6.3 kJ/mol}$,
does it mean that if we divide this number among $\pu{1 mol} \approx \pu{6.022E23}$ molecules, the result is we need $\approx \pu{e-23 kJ}$ for breaking each single hydrogen bond between 2 molecules of water?
It means you need this much energy to break every hydrogen bond in one mole of hydrogen bonds in water, where the number of hydrogen bonds does not have to correspond to the number of water molecules. Every water molecule can form multiple hydrogen bonds; the Wikipedia page оn H-bonding claims about 3.5 H-bonds per water. I am not sure about your value for hydrogen bond energy, the same Wiki page gives about 20 kJ/mol.
