> The MWI literature is festooned with references to "copies" or "versions" of yourself, and "the branching structure of the multiverse".

Many MWI advocates will describe that physical wavefunction in those terms, but that's a metaphor/model/aid to understanding, not a "point of metaphysical fact". The only thing that MWI demands we take as metaphysically real is the wavefunction. Given a wavefunction like 1/sqrt2(|experimenter who observed a live cat> + |experimenter who observed a dead cat>) many people find it convenient to think of this as "two Everett branches with ...", in the same way that people who hear a particular pattern of sound might find it convenient to think of it as a chord of two or three notes. But if you prefer to think of it as simply a unitary (in the everyday sense) wavefunction that's fine, you'll make exactly the same experimental predictions.