I believe the original intention was to tax the market value of the land a structure sits on, not the structure itself. So in the case of apartments, presumably local density would have to be fairly high to make it a good place to build an apartment building there in the first place, and so the market value of the land would also be high; but the units would not be taxed directly at all, though their occupants would have to pay some fraction of the overall LVT. (Presumably that fraction would be higher for the penthouse.)

You can see why the LVT is relatively density-friendly: the taller the building, the less tax paid by each resident.

It corresponds to the value of the unimproved land. So the tax paid by a skyscraper would be roughly the same as that paid by an empty lot of the same size next to it. (You can imagine a system where a tall building would have to pay more tax for blocking out the shared light).

If the burden of the tax were shared equally than the penthouse would contribute the same amount as each of the floors below it—some small fraction of the total for a tall building.

It's taxing fair market value of the land. So, if you erased a skyscraper in downtown, how much would that barren plot of land be worth? Tax all of the value there. It's one of the only known taxes that doesn't negatively distort the economy

in many taxing jurisdictions, property tax is actually a combo tax on (unimproved) land (square footage/acreage) and on the value of the structure. sometimes these are the same rate and sometimes not. in other jurisdictions, market value dominates, though they often consider land value and structural value separately as well. once upon a time i managed a product to help assessors determine building values so they could do the necessary discrimination for property taxes.