My personal hc is that we're in charge of a sort of Purgatory/Land Between where the near dead/spirited away/lost souls are. As my spirituality professor has said "All afterlives are the same; it's the *people* who make it different." Hence anyone, not just Greeks/Romans/Cretians can access the Hotel.
Considering the prologue, it's very possible the MC almost died or actually died from some sort of sleep-deprivation- related accident- we *were* in a train station, after all. We could have fallen on the tracks or something.
Everything with The Old Man (who's real name I'm blanking on) could have been in the "waiting room" of the Styx where shades wait for Charon to transport them. The contract allowed us to "bypass" the typical route and go to what is basically where those who have a near-death/spirited away experiences- the Hotel. Either it was set up as is initially or it became that way from an unexpected interaction of different contracts. After all the dead can't leave Hades nor can you punish the dead, but Asterion is obviously*not* in Hades proper since he's being punished. Ergo, we're somewhere in between where the rules of the dead and living intersect- people just pop up like daisies yet leave just as fast, but people who stay can still grow old, have kids, die, ectera.