You're right, many worlds is the idea that because quantum mechanics involves a particular mathematical way of evolving wave functions in time, measurements "have to" yield all possible results and therefore spawn "many worlds". The multiverse is an idea that our universe is only one part of a larger one with statistically distributed values of the fundamental parameters. So that's not the same thing. But they do share a similar flavor, in that they are both efforts to "explain" without resorting to experiment. In that sense, they both treat an "explanation" as if its job was to generate a sense of cognitive resonance, rather than as a way of organizing data in a way that could lead to predictions for new experiments. In short, they are both efforts to insert what we might call "reality" into a larger milieu that is unconstrained and untested, just to resolve certain "pesky issues" about the reality we find. To me, it makes a lot more sense to simply recognize the pesky issues, than to think of them as in need of resolution using the information we have at hand. Why change reality just to make our minds like it better? It doesn't seem very scientific at all.