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?

I think the fallacies in this argument tend to be different than what you're calling out. After all, $(3,4)$ is an arbitrary point on the dartboard, but some (if not all) of the aspects of the universe called out in fine-tuning arguments (particular masses, coupling constants, etc.) are genuinely non-arbitrary, perhaps even special properties. So the counter is that the dart is closer to hitting a "bulls-eye" (edit: and very often this is an arguable point, but I think it's generally arguable on scientific grounds more than statistical grounds, though the cherry-picking fallacy and survivor bias mentioned in other answers is certainly relevant in some cases).
Where I would say that the fallacy often lies in arguments like this -- and I'd be remiss if I didn't say that sometimes anthropomorphic/fine-tuning arguments can be made rigorously -- is the assumption of an a priori uniform distribution of "how the universe could have been." We have no scientific basis on which to assume that the universe could have been some other way, let alone posit reasonable relative likelihoods for different configurations. Going deeper into why would make this more of a physics discussion than a statistics discussion, but a Bayesian perspective is the appropriate statistical framework in which to view this question (obviously we just have the one universe to look at!).
I should note that some of the above argument is influenced Sabine Hossenfelder's blog, and so her book "Lost in Math" may be relevant here. One of the comments made me go back and re-read the question to see that the argument being referenced in the question is one for intelligent design. I think the above "a priori fallacy" is still present in this case, but so is the survivor bias pointed out in another answer -- there are a lot of fallacies to go around. The assumption that humans take a special place in the universe ignores the fact that other types of universe could have led to other types of self-aware entities. But arguments of similar nature are made even within physics and cosmology, at a level of technical subtlety that may escape accusations of the survivor bias -- e.g. a claim that if such and such parameter were ever so slightly different, no stable atoms could form and the universe would be a featureless bath of radiation. There are a number of underlying parameters that get referenced in these arguments, and since it is beyond my expertise I'll note that the "hierarchy problem" and curiously small dark energy values are two frequent subjects of scrutiny, with conclusions of the arguments frequently being "we need a more natural theory that makes this seem less improbable" or, failing that, "we are part of a multiverse." It's interesting to note that the logic for the second argument is explicitly acknowledging survivor bias as part of its reasoning.
Addendum
To add one last bit of detail regarding why it's a fallacy to assume a uniform prior: I think a lot of people are used to uniform priors being common assumptions in Bayesian inferences, because it seems like the "least biased" thing to do. And a lot of time there's good justification for it, or in any rate the effect of the a priori assumption washes out when evaluating the posterior due to a moderate number of observations. But, especially when there's only one observation to update the posterior, any a priori assumption is a bias. So, that assumption has to be evaluated on its own terms. In the case of a priori assumptions about possible alternate universes, there are a lot of stumbling blocks. For one, the space of possibilities is continuous, without a canonical parametrization, so there's no unique "uniform" prior -- is any value of $X$ equally likely, or any value of $\ln(X)$? We could even learn next year that some "alternative universes" are inconsistent and impossible, i.e. having a probability of strictly zero (one pertinent example in physics, particularly string theory and arguments about its "landscape" or "multiverse", comes from the "swampland conjecture"). For all we know, we'll eventually discover that our universe is in fact unique, in some sense, with all the seemingly random parameters actually the consequence of some deeper structure (a possibly forlorn hope of most physicists), with no leftover free parameters. We lack a complete "theory of everything" that could justify some of these assumptions, and even if we did there would be a philosophical critique: such a theory would be a mathematical extrapolation from empirical evidence, but by definition we cannot collect empirical evidence about universes that don't exist or aren't observable to us.
