The phenomenon of phantom limbs in people who have lost arms or legs to amputation is pretty well-established. It makes a fair amount of sense. The "brain mapping" of the limb (and for that matter the nerves that led to it) don't go away just because the limb does.
But what about people who never had the limb? Does a person who was born with one arm due to a birth defect ever "feel" the second arm that should be there?
Here is a link to an article that discusses the very thing. Specifically, it discusses a case study conducted by Paul McGeoch and V.S. Ramachandran on a woman whose hand was amputated at 18, but who has been missing a thumb and index finger since birth. This woman "sprouted a phantom hand that contained five digits, including a phantom thumb and index finger that had been absent since her birth." The end of the article discusses similarities between this phenomenon and supernumerary phantom limb.
Ultimately (from the article): "The authors say that most people born with congenitally missing limbs do not experience phantom ones, but a small subset do."
In case you are curious, below is the citation for the case study conducted.
As a side note, depending on whose definition you use, "phantom limbs" are defined solely for individuals who have had the limb amputated (though this may be due to unintentionally specific wording).
References
McGeoch, P. D., & Ramachandran, V. S. (2012). The appearance of new phantom fingers post-amputation in a phocomelus. Neurocase, 18(2), 95-97. doi:10.1080/13554794.2011.556128
