I think there are a couple of aspects. One that they have a genetic link to people who control or controlled states. The second element is that they contolled enormous wealth and managed to surround themselves with unparalleled luxury. The fact that this was inherited and not earned added something to the enigma.
Indeed the hereditary part often seems to give monarchs (perhaps especially mediocre or bad ones like NII) some innocence sympathy, when they are contrasted with politicians, who often are politicians because they are good at talking, bargaining, seducing, networking, manipulating etc.
A good example of this can presently be seen in the world's leading monarchy: As contempt of politicians rises in the UK, the Queen gains in respect. When her son and heir occassionally speaks out against
politically correct "modern architecture", some? many? applaud this privileged hereditary royal for speaking common sense and expressing "ordinary people's" views.
Yes, liberal leftwingers are right: Monarchy is not egalitarian. But nowhere, not even in Communist countries, is society totally egalitarian. Because we cannot escape our biological fates. All the things we are born with: parents, gender, looks, brains etc. have an enormous impact on our existence. (So do upbringing and also the biological choices we take during life: Mating, giving birth, parenting or not.) And just like Nicholas II, we have to cope as best we can with these given facts. Even in an age when people don't inherit their farm, trade or lot in life from their parent(s), monarchy still reflects our biological fate much better than a politician as president.
Anybody can become president, but only the kid of the kid of the kid of X the Great can become monarch, you say? Think about it: An average nobody (like Nicholas Romanov or Joe Bloggs) can become head of state in a monarchy by accident of birth, but is not likely to be voted head of state in a republic. Whether it's beneficial to have a Joe Bloggs as head of state is another discussion, but "the silent majority" do seem to find some satisfaction in it, at least only as a revenge upon "the chattering classes".