That's certainly an interesting way to look at it. Usually one doesn't consider the set an object was *selected from* to be an identifying characteristic of the object. The same object can belong to multiple sets, and, mathematically speaking, there is no difference between 2, 2/1, and 2.0. The number 2 is a member of the set of integers, and the set of rationals, and the set of reals, all at the same time. It has a square root in the set of reals, but not in the set of integers. It's a unit in the rings of rationals and reals but not in the ring of integers. The set qualifier goes with the property (has an integer square root, is a unit in the ring of rationals), not the number.
I'm not quite sure what you mean when you say that "rational 2" is not prime... "rational 2" is the same number as "integer 2", which is a prime number. If you're referring to a different kind of prime, like a prime field, then you'll have to be more specific before you can classify 2 as prime or not.
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