You probably find this idea strange and implausible, and I must confess that this is my gut reaction too. Yet it looks like we will just have to live with it, since the simplest and most popular cosmological model today predicts that this person actually exists in a Galaxy about 10^(10^29) meters from here. This does not even assume speculative modern physics, merely that space is infinite and rather uniformly filled with matter as indicated by recent astronomical observations. Your alter ego is simply a prediction of the so-called concordance model of cosmology, which agrees with all current observational evidence and is used as the basis for most calculations and simulations presented at cosmology conferences. In contrast, alternatives such as a fractal universe, a closed universe and a multiply connected universe have been seriously challenged by observations.
The farthest you can observe is the distance that light has been able to travel during the 14 billion years since the big-bang expansion began. The most distant visible objects are now about 4 x 10^26 meters away, and a sphere of this radius defines our observable universe, also called our Hubble volume, our horizon volume or simply our universe. Likewise, the universe of your above-mentioned twin is a sphere of the same size centered over there, none of which we can see or have any causal contact with yet. This is the simplest (but far from the only) example of parallel universes.
By this very definition of "universe'', one might expect the notion that our observable universe is merely a small part of a larger "multiverse'' to be forever in the domain of metaphysics. Yet the epistemological borderline between physics and metaphysics is defined by whether a theory is experimentally testable, not by whether it is weird or involves unobservable entities. Technology-powered experimental breakthroughs have therefore expanded the frontiers of physics to incorporate ever more abstract (and at the time counterintuitive) concepts such as a round rotating Earth, an electromagnetic field, time slowdown at high speeds, quantum superpositions, curved space and black holes. As reviewed in this article, it is becoming increasingly clear that multiverse models grounded in modern physics can in fact be empirically testable, predictive and falsifiable. Indeed, as many as four distinct types of parallel universes have been discussed in the recent scientific literature, so that the key question is not whether there is a multiverse (since Level I is rather uncontroversial), but rather how many levels it has. [...]»
(Max Tegmark, "Parallel Universes", [astro-ph/0302131])