Intramolecular bonds are quite a bit smaller than intermolecular distances, so if you pack atoms into longer molecules you get a denser result.

I'm guessing very long molecules become problematic (because they don't pack well), but ammonia is a small molecule so way below that.

Liquid ammonia has a density of ~690 kg/m3 and a molar mass of 17 g/mol so ~40k mol / m3, as NH3 that's ~120k atoms of hydrogen per cubic meter. Meanwhile liquid hydrogen has a density of 71g/L and a molar mass of 2.02 g/mol so a very similar ~35k mol/m3 but as H2 that's only ~70k atoms of hydrogen per cubic meter.

And liquid hydrogen aside from being extremely flammable, can't exist above 30K and degrades storage material (https://en.wikipedia.org/wiki/Hydrogen_embrittlement), ammonia is much more forgiving and liquid at ambient temperature above 1MPa (10 times atmospheric pressure), not innocuous by any means but way easier to transport and store (storage requirements are similar to propane).

Probably also because ammonia is polar, three positive protons and an electron pair in a tetrahedral arrangement. Whereas H2 is two protons hiding an electron pair in a linear arrangement.

spot on

~120k mol/m^3 and ~70k mol/m^3

Armchair chemistry ahead:

Molecular hydrogen is unpolarized while ammonia molecules can establish hydrogen bonds, thus packing them more densely under the same conditions.

For really dense hydrogen we would need its metallic form but that requires pressures you might find in the core of jupiter. The electromagnetic force is more convenient than gravity.

I would expect this is due to the fact that each ammonia molecule contains three hydrogen atoms, whereas a hydrogen molecule contains only two. So liquid ammonia would be expected to contain more hydrogen per volume than liquid hydrogen.

That would be true of their gases.

Wouldn't it be true of both liquid and gas?

Ammonia is 10x more dense than hydrogen gas at the same temperature/pressure, so you can pack more of it into the same volume.

"liquid hydrogen"

It applies to liquids as well.