I do not believe they are arguing for a soul but rather just pragmatism. It's not that “measurement” has some special status as coming from a soul or something: indeed your unwillingness to see it as a normal thing that happens, such that you immediately jump to this question, indicates that you have already “drunk the Kool-Aid.”

This is more clear in Everett's original formulation. Everett didn't speak of “many worlds” but of a difference between relative and absolute truth. You measure a spin-½ particle along some axis, it is only “relatively” true that you saw what you saw, say ↑, and there is also a relative truth that you saw ↓. But because these relative truths are exhaustive there is also an absolute truth that you have the deluded belief that you saw either one or the other; that truth holds in both relative truths. The “lost contact” objection is precisely that in Everett's theory this belief is ultimately delusional (there is nothing like collapse to which they correspond, it is just formally incorrect to confuse your relative truths with the absolute truths) and we are led by the theory to delusions like this “with high probability” (scare quotes because Everett realized more so than his successors that eliminating collapse also untethers the theory from probability in a deep way). It is just accepted that these deeply practical things like “my existence as an observer” and “my tool’s measurements” are ultimately based on a sort of illusion which has no correspondence to the true reality; this is buried in a mathematical technicality in his thesis (he notes that he is not looking for an “isomorphism” between experience and the external world, but a “homomorphism”), but it is kind of the crux of the whole enterprise. “It’s fine if I predict things which are not observed, so long as I also predict the things that are observed and I predict that you will be very opinionated about not observing the things predicted but not observed,” if you will, is the homomorphic approach to QM that Everett advocates.

In that respect there is something very different from “the history of physics.” Physics does in some cases say that certain things are illusions. And those statements track very closely with MWI. For example a rainbow appears to be a thing out in the world, but modern physics is very happy to say that the phenomenon cannot be correctly located as an object inside the cloud in which it is perceived, because it turns out the cloud is “rainbowing” in different ways in many different directions and you are only getting part of the story. You see part of the light suggesting a reconstruction of a physical object, but if we look at all of the light we realize that the different reconstructions are not reconcilable because they all have to occur at a 41° angle from the rays of the Sun and real objects have varying angles.

You can see a lot of similar ground trod here. Both claims of illusion appeal to the act of trying to reconstruct the external world from observed experience. Both then appeal to looking at all observations taken together. But there is a difference in scope. In the rainbow case there is a counterfactual, a “what you would have seen,” that can be confirmed with a camera at some other location. We can do a parallax measurement to reconstruct that the rainbow is actually around as far away from you as the Earth is from the Sun or so. But in MWI these sorts of experiments which may be possible are ultimately forever infeasible, I need to have quantum control of every atom of my measurement apparatus if I want to make some similar observation, and even then any particular observation will be consistent with a classical ontology, it's just the pattern of many observations which will suggest non-local correlations which I can interpret as evidence for maybe everything being illusory. One rocks your boat gently, the other is sailing in a hurricane.