Obviously, "the bogeyman" is fairly amorphous, so what I'm referring to here is a mysterious, terrifying entity that parents used to scare their kids into compliance. Or, a mysterious, terrifying entity that people used to explain odd, inexplicable phenomena. Or both.
Where I come from (Southern California), our "bogeyman" is the chupacabra, but I imagine different areas of the West have different versions, and I want to know how old and universal these are.
Do we have ancient/classical texts referring to similar entities (fantastical horrors that parents used to frighten their children), and what do those look like?
Aristophanes referenced Mormo in his plays The Archanians and Peace. Mormo would bite bad children. It would be difficult to name all of the spirits, monsters, etc. that target misbehaving children, there are simply too many. But there is a long history.
It's worth noting that there was no line drawn between myth and reality in the ancient world. These things were real to them.
Do we have ancient/classical texts referring to similar entities (fantastical horrors that parents used to frighten their children), and what do those look like?
Here's the thing: belief in malevolent supernatural beings could indeed serve social functions, but that doesn't mean that people didn't actually believe in them. There were certainly some skeptics, but the Pre-Modern imagination populated the forest, wasteland, bog and other fringes of civilization with kinds of supernatural creatures and spirits. 'Boogeymen' like hobgoblins or dracs (I think this is the correct term but I could have forgotten) could even inhabit households and urban areas and also served social functions of frightening adults into working and cleaning industriously and keeping careful watch over their children.
In Britain and Ireland, fairy belief definitely served social functions similar to the modern boogeyman - the fairies kept people from wandering into the woods or near bodies of water at dusk or dawn, made parents keep vigil over their infants, made housewives work industriously etc. But this was all predicated on popular belief in the fairies. Pre-Modern boogeymen were legitimately believed to have existed by the majority of the population, and weren't just cynical inventions devised to keep children and women in check. Their power was derived from their hold on the imagination.
So I think you should approach the issue this way: you have to recognize that Pre-Modern people had a vastly different worldview and that both adults and children sincerely believed in boogeymen. They served an anthropological, social function, but they were still genuinely believed in by the majority of the population.