Yes @APearOfPearsThatsHowIRoll the vomiting dog got me too! Wtf? OK it's an app, not a great work of fiction, but also not a sleep-deprived conversation where words and ideas just aren't working for you, surely at some point they could have thought of a less weird example than that?!
In general - hmm. The idea of leaps was vaguely around when I had my first in 2012 (I remember downloading a couple of chapters on my kindle), and seems very widespread now with my second (15 days off leap 6 apparently!). First was a grumpy git anyway. Second seems to have followed the timeline fairly closely - could be total confirmation bias - I glance at it from time to time with the same curiosity I did 'what animal is my unborn baby the same size as this week' when she was in utero.
It makes sense to me that babies can become more clingy and crappier at sleeping when they're working on a 'leap' in development. As something quite concrete, I remember my firstborn (always an awful sleeper) becoming even more awful when he was learning to crawl, because he kept fucking crawling in his sleep, head first into the wall, grunting in annoyance. In my (sample size very small) experience, child development does seem to absolutely come in fits and starts, rather than consistent gradual increments. So some of the concept is defensible. Risible to suggest it happens on such a precise and universal timeline, yes.
At the same time though I do think there is something to it which is slightly more than confirmation bias, and this helps people buy into the idea - I think its no coincidence that one of the leaps (four?) coincides with the also-cringily-talked-about "four month sleep regression". Because this is totally a recognisable phenomenon, right - at about that age, sleepy newborns wake up to the world, start understanding a whole lot of stuff, are easily distracted, hard to settle, easily awoken, and grumpy as sin because that's a fairly rubbish thing to adjust to. Not ALL babies, sure - but it's a thing, isn't it? If 'four month sleep regression' or 'leap four' helps parents quickly convey that in conversation with peers, or feel better and remember that it's a phase, or whatever, I don't really see the harm.
I am guilty btw of engaging in 'how has your baby been in leap x' chatter. I've read just enough on it that I can join in, and for all I do wince slightly sometimes, I think I'd wince all the more at someone pointing out how crap the evidence base is. I wonder if lots of these mums are like me in that, rather than fully bought into the idea?
Also - my mum often comes out with this "in my day we had none of these books and stupid theories and blah blah" stuff. I'm actually not sure that's strictly true, or indeed if it's actually preferable if it was. I totally see that a whole industry has sprung up to exploit new parents' anxieties and some of it is downright mad, but like faddy diets I think it has a longer history in some form or another than people are willing to admit or remember. Plenty of the hippydippy attachment/continuum parenting writing goes back to the 1980s (1970s?), doesn't it? Never mind Winnicott or Spock. Penelope Leach. Parenting 'experts' are not such a recent invention, and mindlessly copying what your mother or sister or neighbor did isn't necessarily any better than dutifully following the baby whisperer.