But in the MWI what we start with is the universal wavefunction |Psi>. Say there are 10^80 particles in the universe and they may have a spin and could be in infinite locations. That makes for an ininite-dimensional Hilbert space. If we consider that the precision in the location variables is limited by Planck's length there are still around 10^180 possible locations in the universe for each particle. The Hilbert space where the quantum state of the universe is defined is huge.
Now, you say "the state looks like |photon>⊗|me>". To keep things simple and in the realm of Schroedinger's equation I will consider an electron instead of a photon, prepared in the state spin z=+1/2.
Is "|up>⊗|me>" shorthand for "(1 |up> + 0 |down>)⊗|me, the electron at that position, the rest of the universe>" which would be the "in-branch" quantum state of the universe?
If indeed |Psi> can be written as a superposition of that state and an inconceivable huge number of orthogonal states, are all those states (when their coefficient is non-zero) the multiple worlds?
But given |Psi> there are infinitely many (or at least a lot of) possible basis to express the pure state as a decomposition (including of course the one where the only non-zero "world" in the superposition is the one described by the wave function |Psi>). Do we count as "worlds" as well the states appearing in other basis?
Is this slicing of |Psi> into other states just a mathematical trick or does it have a physical meaning? Even if we can redo the operation at different times on the evolving |Psi(t)>, how are the "emergent substates" at different times related?
The answer to all these questions is far from trivial.