The many worlds interpretation holds that physical reality is described by the formalism of quantum mechanics without the collapse postulate. The theory includes a mapping between the terms invoked in equations and the sort of thing it is possible for us to observe. Once you have put this in place and applied information theory to quantum mechanics, the information theoretic uncertainty principle follows. There is nothing particularly unusual about this.
There is something very unusual about the discussion over the interpretation of quantum mechanics as David Deutsch has noted here and here. The problem is the following. Any scientific theory is not just a bunch of predictions. Rather, it is an explanation of how the world works and the whole idea of making predictions makes no sense unless that is the case. The way you do any measurement is set up a physical system that you hope will record the results of doing some particular thing to some physical system. There has be something happening in the real world to make the results of the measurement have a correspondence to what you are trying to measure. And attempts to improve measurements have to consist in inventing better explanations of how to produce such a correspondence. Predictions and explanations are not independent. As a result you can't divorce predictions from the picture that your theory gives of how the world works.
Quantum theory without collapse predicts the existence of multiple versions of the system being measured. This is not an optional extra. It is not something you can toss on a whim if it happens to suit your mood. You ask how an interpretation can make predictions not made by the other interpretations. Let's look at some of the alternatives. The Copenhagen interpretation amounts to sometimes quantum mechanics applies and sometimes it doesn't but we're not going to provide a precise explanation of why and how: this makes no predictions at all. Hidden variable theories: quantum mechanics is false all observables are always single valued but this happens to reproduce the predictions of quantum mechanics in some completely unspecified way. Since this doesn't explain how and where the hidden variables differ from quantum mechanics, it also makes no predictions. So the MWI makes more predictions than both of those theories.
Note also that the denial of the existence of the multiverse has helped lead to bottomless pits of confusion about quantum mechanics, such as the idea that quantum mechanics is non-local. In EPR experiments after the measurement is done the measuring instrument and the systems that carry the measurement results exist in multiple versions. Those versions only become correlated after they are compared and this process is entirely local. This happens because the measuring instrument and any other system that carries the measurement results contain locally inaccessible information: their observables depend on what measurement was conducted, but the expectation values of those observables don't have that dependence:
There is no way even to express this explanation if you are not clear about whether or not "classical" systems, that is decoherent systems, are described by quantum observables and whether they exist in multiple versions.
Variants of the MWI may or may not make the same predictions. Some of them are not clear about what exists in reality. For example, consistent histories people often fudge the issue of the existence of the multiverse and I find it very difficult to tell what happens in reality in Zurek's existential interpretation. The many minds interpretation claims that the slicing of the multiverse into parallel universes is dependent on the existence of minds somehow. I don't think this is the case since decoherence can take place without minds and it suppresses interference. There are cases, such as the EPR experiment, where the parallel universe approximation breaks down for macroscopic objects, but this seems to happen whether or not people are present, so I think the many minds theories are false.
The best elaborated versions of the MWI are the Heisenberg picture based version by David Deutsch
and the spacetime state theory by Wallace and Timpson
You might also argue that the Montevideo interpretation, which takes account of decoherence caused by clocks, is a version of the MWI:
As far as these theories are concerned I think we're beginning to get into research areas where it is not clear to anybody how to work out the correct predictions so we don't know how things will turn out.
As for other interpretations making predictions, the ones that do make predictions are alternative physical theories that are not the same as quantum mechanics, such as the GRW theory.