I am of two minds with respect to this question; therefore I will lay out the case for the existence of prime factorizations of positive rationals such as $2.5$ or $\frac{22}{7}$, and then I will lay out the case for their nonexistence.
The prime factorization of any natural number $n \ge 2$ is a product of its prime factors such that the product evaluates to $n$; we are guaranteed by the fundamental theorem of arithmetic that the product is unique up to the order of the factors. Thus we may write $40$ as $2 \times 2 \times 2 \times 5$, or more briefly in exponential form as $2^3 \cdot 5^1$. Let us look at the generalized exponential form of the prime factorization of $n$:
$n = p_{a_1}^{b_1} \cdots p_{a_k}^{b_k}$,
where $p_{a_{i}}$ is the $i$th prime factor of $n$, $p_{a_{k}}$ is the greatest prime factor of $n$, and $(b_{1}, \ldots b_{k})$ is the unique sequence of positive integers satisfying the equation.
But let us now relax our requirement that the integers $b_{i}$ in $(b_{i})_{i=1}^{k}$ be positive, allowing negative integers to be included as well. Then we could write $5.6$, for example, as
$2^2 \cdot 5^{-1} \cdot 7^{1}$.
Observe that the above expression bears a striking resemblance to the exponential form of prime factorization. Indeed, just as with prime factorization, the expression is unique up to the order of its factors, since $5.6$ can be uniquely expressed as the reduced fraction $\frac{28}{5}$, the numerator and denominator each corresponding to unique prime factorizations:
$5.6 = \frac{28}{5} = \frac{2\times 2 \times 7}{5}$.
On the other hand, the fundamental theorem of arithmetic only addresses integers greater than 1. Also, prime factorizations are frequently written without use of exponentiation (as I illustrated with $40$ above); this is impossible if negative exponents are permitted. For instance, there is no integer $k$ satisfying
$5.6 = 2 \times 2 \times \underbrace{5 \times \cdots \times 5}_{k \text{ $5$s}} \times 7$.
So in one respect it appears we can generalize prime factorization so that every positive rational number $q$ has a prime factorization (the prime factorization of 1 being vacuously 1 itself), yet in another respect it appears such a generalization is not possible.
Can we speak generally of "the prime factorizations of positive rationals," then, or not? And if not, what would be the proper term denoting such objects?