That having been said, what is it that makes these particular values "prime"? It's that they cannot be (nontrivially) decomposed into further factors. But in the rationals, they can be! 2 = 1.6 * 1.25, 3 = 1.6 * 1.875, etc. The inability to further factor 2, 3, 5, 7, 11, etc., which causes us to single them out as "prime" in the first place, is only on an account of factorization into whole numbers excluding such rational factorizations.
So it's in that context that my comment was to be understood: the fact that 6 specifically has the irreducible factorization 2 x 3 is not universally true, but only true in particular contexts, of which the most common would be the integers. In the mentioned context of the reals, or even just the rationals, we could instead note that 6 (or anything, really) could continue being divided indefinitely, thus having no particular "prime factorization" in the sense of a factorization which cannot be further refined.
Not quite. You can easily extend prime factorization to rational numbers by allowing negative as well as nonnegative exponents for primes. The prime factorization for 6 (and any other rational) will still be unique.
If you instead decided to represent the factorization of a rational as a list of factors for the numerator and another list for the denominator, your factorizations would stop being unique... but the first way works just as well.