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?
 A: 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.
A: The Fallacy is not statistical. It is related to determinism. old science was deterministic and also stiff . So the butterfly effect was a logical consequence: you change something just a little bit and the future is completely different as a result.
This is not how science is understood today. Determinism is gone, and even In stiff systems we are finding attractors. In other words, today we should easily envision that small changes do not matter. You step on a butterfly in the past, come back to today and see the world is indistinguishable the same. OR you go back to past , don’t kill a butterfly but when you come back the world is completely different because it is not deterministic. Every time you run it, you get a different result.
The world is robust to disturbances, at least to some of them, because it is also non deterministic
A: The universe and mankind
The fact that we observe the unlikely event of a universe, solar system and planet that is able generate intelligent life, is a type of survival bias.
(and as Mehmet mentions in the comments, this could be seen as a cherry picking fallacy)
We see the unlikely event because without it we wouldn't have lived to see the absence of the event.
In relationship with the existence of life this phenomenon relates to the anthropic principle. This has different forms. From Barrow and Tipler's book:

Weak anthropic principle (WAP): 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.

The world and universe are not very probable, but it is the way it is because there is a selection effect.

Strong anthropic principle (SAP): The Universe must have those properties which allow life to develop within it at some stage in its history.

In this case, the properties of the universe are not regarded as probability but as some sort of intelligent design (this is also more like a philosophical argument, involving teleological ideas, and may not be so much of a fallacy). But it can be even adapted further. With interpretations of quantum dynamics, it is not just that the special unlikely conditions for the universe are necessary for humans/observes, but also the other way around observers are necessary for the universe to exist.

The zero probability

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)".

This argument is slightly different from the survival bias. The argument is that events with zero probability are not supposed to happen.
This is misrepresenting 'probability' (which we could see as an equivocation fallacy).
While the result/outcome 'the dart hits (3,4)' is part of the sample space, the set of all possible outcomes (often denoted as $\Omega$), it is not in the event space. The result/outcome 'the dart hits (3,4)' is not an event in event space to which we can formally assign a probability.
The situation is a little bit similar to Zeno's paradox. In a similar sense as Zeno's argument for motion not being able to occur, we could argue that the arrow can't hit any part of the dartboard because the probability to hit it is zero everywhere.
A: 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.
