Is there a formal name for this statistical fallacy? "Given the infinite amount of possible configuration of the universe and the fact that if something in the universe were slightly different, we wouldn't exist, we can conclude that it couldn't (or probably could not) be created by the randomness".
It's such a common argument when someone tries to argue the necessity of a rational and active agent creator for the universe that it we see it in an Sheldon's series episode. But is not hard to notice that it's a statistical fallacy exactly like that one:
Suppose I threw randomly a dart in a circle and it hit exactly at the point (3, 4), if one says for me: "There infinity many points you could hit, so, if you threw it randomly, the probability (using the limit definition) of you hitting that point is 0, therefore you didn't throw it randomly and must have something external being that rationally made this dart hit (3,4)".
In this kind of fallacy we assume that a certain state in a sample space should have a high probability to be random because it's arbitrarily labeled as "special", but, in any continuous distribution, by definition, all possible states would have a probability equals 0.
Is that a known fallacy in the formal stats world? If so, does it have a name?

Statistics: Texas sharpshooter fallacy
The fallacy you describe resembles the Texas sharpshooter fallacy, in which one reverses the order of sampling data and fitting one's hypothesis so that the observation confirms it. From the linked Wikipedia article:

The name comes from a joke about a Texan who fires some gunshots at the side of a barn, then paints a shooting target centered on the tightest cluster of hits and claims to be a sharpshooter.

Beyond: Anthropic principle
On a deeper level, a response to the reasoning from finetuning comes with the Anthropic principle, whose core idea could be summarized like:

The observed values of all physical and cosmological quantities are not equally probable but they take on values restricted by the requirement that there exist sites where carbon-based life can evolve and by the requirements that the universe be old enough for it to have already done so.

In other words: we should not be surprised to find ourselves within an apparently fine-tuned universe, because otherwise we just could not have come into existence. Several variants, loosely grouped into "weak" and "strong" anthropic principle, exist, as Sextus Empiricus' answer nicely summarises.
