Imagine infinite set of, let's say, natural numbers. I choose one of the infinite numbers randomly. Let's call that number n. If I choose another number too, can it be the same number (n), theoretically?
As mentioned in the comments, there is no way that each natural can have the same probability. See this answer to "Can you pick a random natural number?..., for instance.
However, we can still pick a number randomly with unequal probabilities. For instance, suppose I choose a number by flipping a coin, and the number is the number of heads before the first tail. (If you want to include a case of flipping heads forever, let's say that also counts as the number $0$.)
I just got the following sequences of flips: $HHT,T,HT,T$. Which means that the number $0$ was randomly selected twice.