I’ve been struggling a bit with prime divisibility and algebra, is my understanding of this correct:

if given xy^2 = 180, you should prime factor if to get 2^2.5.3^2. Since y is squared, the only option for it is 6 (6x6 = 36), and then x would be 5? Or are there other ways to solve?

First of all, we must know that x and y are integers and can’t be fractions or irrational numbers, otherwise, the number of values would become infinite

No, that would just be one of the possibilities, the only thing we know is that 5 can never be the factor of y as it is not squared
Thus, all other combinations of 2 and 3 are viable:
y = 1, 2, 3, 6
x = 180, 45, 20, 5