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John Wheeler proposed along with Hugh Everett and Bryce DeWitt the 'Many Worlds' Interpretation of Quantum Mechanics (although he expressed some doubts about its validity)

I have read, from an article in Nature, written by Martin Rees and Bernard Carr back in 1979 (https://www.nature.com/articles/278605a0) that Wheeler proposed some kind of multiverse containing universes with fundamentally different laws of physics:

There may already be room for the anthropic principle in this picture. Wheeler envisages an infinite ensemble of universes, all with different coupling constants and so on. Most are 'stillborn', in that the prevailing physical laws do not allow anything interesting to happen in them; only those which start off with the right constants can ever become 'aware of themselves'. One would have achieved something if one could show that any cognisable universe had to possess some features in common with our universe. Such an ensemble of universes could exist in the same sort of space as the Everett picture invokes.

I have also contacted with science historian Helge Kragh who in his book "Higher Speculations: Grand Theories and Failed Revolutions in Physics and Cosmology" he wrote about this as well:

Nozick 1981, p. 129. The fecundity assumption is related to the principle of plenitude, but the latter refers only to the realization of possibilities in the actual world we live in. Whereas Nozick did not mention the anthropic principle, he did refer to Wheeler’s speculations of many universes with different laws of physics

And he told me that Nozick referred to "Everett-Wheeler's Interpretation of Quantum Mechanics"

But how can this be true? Why did Nozick refer to Everett and Wheeler's many worlds interpretation if their universes would not have different fundamental laws?

I mean according to this interpretation all universes would have the same fundamental laws, so all of this does not make much sense to me.

So, how can all of this be compatible? Did Wheeler propose any kind or alternative version to the "traditional" MWI where universes would have fundamental different laws and rules?

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Not exactly. First, the different branches of the universal wavefunction (under MWI or any other interpretation) are different from the "universes" of a multiverse model.

Wheeler's cited comments (from Patton & Wheeler, Quantum Gravity, Ch. 9) hypothesize a situation where the various coupling constants arise in quantum mechanics during cosmogony, and thus different branches of the wavefunction get different values. This isn't "some kind of variation of MWI" as much as just MWI applied to a particular form of quantum gravity.

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