If cosmological objects similar to the Earth and the Moon were observed orbiting a star other than the Sun, they would be classed as a 'Dual Planet' - i.e. two planets which orbit each other as well as orbiting the host star. The large size of the Moon (~1/4 the diameter of Earth) gives it the Dual Planet status. (Other known moons are generally far smaller than the planet which they orbit.)
In addition, through the collection of rock samples from the Moon (and other methodologies) it's now known that the balances of elemental isotopes in Moon and Earth rocks are so close as to be considered identical. And yet very distinct from other planets in the Solar System. Further supporting the Dual Planet idea.
It's thought therefore, that either the Earth and the Moon coalesced (via gravity) from similar cosmic debris at the same time and in the same region of space, or, that one larger coalesced body somehow split into two. (Note that the process by whichis itself not yet fully understood).
The current leading theory is that the early Earth was partially impacted by a Mars-sized rogue planet, which has been named Theia, throwing enough material into orbit around the Earth as to eventually form the Moon through gravitational coalescence. Although this is the dominant hypothesis, it has been widely challenged. Principally because such a huge impact would have left very substantial quantities of Theia material in the mix. Since Theia, like other planets, would presumably have had very different isotope balances, this would be easily detectable in the mix. Analyses show that it's not.
(Note : Some theorists suggest that the remains of Theia may have been incorporated deep in the molten rock part of the Earth's mantle. See :)
There are many other plausible theories regarding the formation of the Moon. Many of which, like the Theia theory, have been confirmed as being viable using computer simulation models. It may, for example have split away from a rapidly revolving molten Earth simply due to angular momentum. Or, there could have been a nuclear-based detonation within the Earth's core.
For details of these and other theories, see : Wikipedia
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