The other answer has already shown that this is not the case.
However, it is also true that: "If we take random samples from a continuous distribution, the probability of repetition is $0$"
We can identify the mistakes that make the conclusion drawn from the above statement invalid
1. What is repeating?:
Note that we can only say that the probability of a repeat of the exact same observation is zero - and not the probability of some random observation from a large class of similar samples.
Recall that humanity in particular, and life on Earth in general do not contain all possible life. Rather we are a very particular case of the infinite (in the sense of infinitely many variations within the set) possibilities of life.
Thus, while we may say that the probability of finding a planet identical to Earth with life exactly like us on it elsewhere in the universe is $0$. This does not mean that the probability of finding life itself - intelligent or otherwise - is $0$. Life is like an interval, not like a point.
In other words, we may not find another Omar Adel asking this question on Physics.SE, but we do find other users asking physics questions on the Internet.
2. Probability 0 $\neq$ Impossible:
This may seem counter-intuitive, but it is the case in certain contexts - including this one.
For example, consider a number line or a Cartesian plane. Because there are infinitely many points, the probability of choosing any given point at random is $0$. However, if we were to run a trail, we will choose some point, even though a priori, the probability of choosing it was $0$. (This would also hold in the case of countable infinities, such as choosing a random natural number. However, note that such distributions would not be continuous - though they can be uniform - and are not relevant here).
Thus, in the context of such continuous distributions, probability zero may not imply impossibility. Indeed, the concept of measure is more suitable here.
Thus, the fact that the probability of a repeat is zero, does not mean that the existence of a copy is impossible.
3. The universe:
More general than the other two critiques is the fact that correspondence between the distribution for which the result holds and the universe is far from exact.
Leaving aside the fact that we currently do not think that the universe is infinite, it is easy to imagine infinite-universes that are counter-examples to this.
Imagine an infinite universe that is empty except for 2 (or more) identical particles rotating around their common center of mass. Here, the variety of structures is not infinite, rather there is either empty space or a particle - and both configurations repeat at least twice!
In fact, many qualitative arguments about infinite universes (including the one in CR Drost's excellent answer) tacitly assume some non-trivial properties that need to be fleshed out before we can say anything that is so mathematically precise. After all, what stops an infinite universe from simply being 'tiled' by a uniform grid of identical atoms?
Thus, we need to make many assumptions before the argument even applies to an arbitrary infinite universe.