this is vaguely related to the string theory related post from a day or two ago, it's all bazinga science folks TL;DW string theory is a big thing because people that read pop science really liked it and it took a long time for physicists to come out in force and say "this is untestable garbage"
Isn't that part of the most popular account of quantum probability?
Not really, no. It's probably not right to call the Everett interpretation the most popular account of anything; it's got a niche, but it's not super popular with most physicists. Even setting that aside, calling the Everett interpretation "many worlds" or "multiverse" is kind of a misnomer. The wave function has a branching structure with respect to certain events, but even on distinct "branches," you're not really looking at separate worlds in the standard sense. For one thing, branches can recohere on the Everett interpretation--it just requires a very particular series of events.
This is always something that's bugged me about the popular depiction of "multiverse" stuff as well (think Sliders, Rick & Morty, etc.): if you can travel between "universes," in what sense are the multiple universes instead of just one universe with a very strange geometry? It seems like if causation and human beings can move between two points, then those two points are by definition in the same universe.
What is the most popular interpretation then?
Among physicists, probably still something like Copenhagen. Most of them don't really think about it too much.
I vaguely remember seeing a study that polled physicists and found that most of them supported copenhagen. It helps that the copenhagen interpretation is kind of the bare minimum you need to do quantum mechanics and the issues with it like what counts as a measurement never come up in practice: a measurement is whenever the physicist measures something.
Copenhagen.
The whole pop-culture "multiverse" trope is also not taking into account the fact that only quantum particles are able to exist in an uncollapsed state. The thing that collapses a wave function isn't magical observation, it's a concentration of quantum particles interacting with each other.
A human sized collection of particles is incredibly stable because of the "inertia" of those particles interacting with one another. The Schrodinger paradox wasn't meant to be some big revelation about the nature of quantum systems, but a joke about how we don't really understand yet what it is that causes the collapse of the wave function and viewing quantum events outside the context of the local systems they occur in is bound to lead to paradoxical conclusions.
But we get multiverse pop-culture stuff because it's an easy cop out for writers.