Actually, those are two separate hypotheses. Hypothesis 1 is that there is an infinite number of worlds. Hypothesis 2 is that everybody of us exists in an infinite number in infinite versions in infinite worlds.
Hypothesis 1 is a much weaker hypothesis than hypothesis 2. If we replace “infinite” with “astronomically many”, then many cosmological models say that hypothesis 1 should be true; including eternal inflation, an infinite spacial size of our universe, and the multiverse interpretation of quantum mechanics.
Hypothesis 2 is much stronger and is in particular implied by modal realism and its versions. My own philosophy states that the world is effectively the totality of all mathematical structures, which also includes all possibly physical worlds (strictly speaking those are two different statements which could be true or false independently from each other).
If hypothesis 1 turns out to be true, this doesn’t mean that hypothesis 2 is automatically true. It could very well be the case that the world is infinite in extent, but that it is structured in a way that stops that from producing all possible variations of individuals and individual-world relations. Yet, that seems very unlikely. Just worth pointing out that this case could be possible, still.
Good that you point that out. This is a complication that is caused by existence of different ways of defining identity. Let’s start by comparing the objective view with the subjective view.
The objective view of identity
It’s hard to define identity objectively. Humans are system embedded in a much larger system that we call the “world” or the “universe” or the “cosmos”. Furthermore, humans are no static systems. They change over time. It may therefore be more appropriate to see humans (in the context of them being beings with a history) as system-valued functions. At their birth a human is a system h(0). At their first birthday that human may be a system h(1). If that human dies at age 75, they will then be a system h(75). In this sense, the human in question can be seen as the whole function h that is defined on the interval [0,75] (actually [-0.75,75] if you consider the typical time before birth). At each point t in that time, the function h provides a value h(t) that completely defines the properties of the human in question. A very reductionist approach to that would be to take the properties of all elementary particles that the human consists of and taking that set of properties to be h(t). Of course, that would violate the Heisenberg uncertainty principle, but let’s just ignore that for a minute a claim that we can nevertheless define and measure all those properties at the same time with absolute precision anyway, just for the sake of argument. There are certainly more meaningful ways to define an individual, but this way is at least conceptually one of the simplest possibilities to do that.
Now, according to hypothesis 2, h is defined in an infinite number of universes. Let’s consider two universes for a start in which that is the case: universe N and universe S. Universe N which is not a simulation, or alternatively is a simulation in which the simulators don’t bother to resurrect h after t=75. In universe S h is simulated in a world S*, but the simulators decide to resurrect h after their death in the simulation. In mathematical terms this means defining a continuation of the function h that extends beyond t=75. It may be the case that this continuation lasts until t=1 000 000 or so, or until the heat death of universe S*, or infinity, if S* happens to be some kind of strange universe that lasts forever without degrading in any way.
From this objective view we now have two functions: h_N, which is strictly defined in the interval [0,75] and h_S, which has an extension, let’s call it h_S*, that is defined on a much larger interval than [0,75].
We can of course see that h_N and h_S are two different functions, because they are defined in relation to different universes in which they are defined. Yet, from a purely mathematically perspective, both functions are the same, because they have the same interval on which they are defined, and because they have the same values in those intervals. For all practical purposes the persons modeled by h_N and h_S are the same. The only difference is that we say that h_S has a continuation h_S*, but h_N doesn’t.
The subjective view of identity
It’s even harder to define identity subjectively. As subject you don’t experience yourself as configuration of particles, but as stream of subjective experiences. We currently don’t know how to model subjective experiences mathematically, or whether that’s possible at all (I think it is, but it’s probably very difficult). We even don’t know how subjective experiences relate to objective configurations of particles. This is of course the old mind-body problem.
But what’s relatively clear is that in our subjective steam of consciousness we have no idea what kind of world we are embedded in. It could be universe N or universe S, or anything else. So, we have no idea whether we will have a continuation of our subjective experience or not.
However, we cannot experience our own nonexistence. That would simply be impossible. Therefore, the only thing we would be able to perceive after a death in a simulated world, would be our resurrection in the world in which we are simulated in. In that sense, we are subjectively immortal, because we cannot experience permanent death, but only subjective continued existence in the cases that the simulators continue our existence.