So, in short, no, two distributions will not be the same "specifically because they are random". They can be, though, but that would be despite they are random, and that would be an extremely unlikely event
Okay, let me try to explain it
It is not like you will have the same outcomes, and the distributions thus obtained won't be the same like two exact copies of something. Think of it as metrics that allow you to determine and establish with a certain degree of certainty that you are dealing with a random distribution and which should remain the same within a specified range
If these metrics do not follow a set of those describing a random distribution, the distribution you are analyzing is not random. It is like an accounting identity or an equation. If you know one part or side of it, it is irrelevant what makes up the other as its result will still be the same, no matter how many elements it may contain or what functions it may have