Based on the comments suggesting that this was used to describe the number of species of a certain family of insects, or something similar, I would say that this is a perfectly correct prosaic use of the term infinite.
This misuse of the word "infinite" alludes to the fact that there are many many many insects in that family. Much more than we can imagine. This is very similar to how we say that solar power is unlimited power, and that the internet has an infinite supply of pictures of cats.
But note that this is indeed not a mathematical context. In a mathematical context something is finite or infinite, but not both. Especially since the modern definition of infinite is "not finite".
"If you had an infinite list of numbers, it would clearly contain every number, right? . . . Because a truly infinite list would contain all numbers."
Certainly not.
For instance, the list "$2, 4, 6, 8, ...$" doesn't even contain every natural number! (Note that we are indeed talking about infinite lists here.)
So that's the mistake you're making; now, why is Cantor's claim true?
You ask how we can produce a number not on a given infinite list; well, this is what Cantor does! For simplicity, let's look at real numbers in $[0, 1]$ in base $2$, and temporarily ignore the fact that some reals like $0.0111111...=0.1000000...$ have multiple binary expansions (this is easily fixed later on, but makes things harder to understand at the beginning).
Now suppose I have a list of these reals - that is, a sequence $r_i$ (for $i\in\mathbb{N}$) where each $r_i$ is some real in $[0, 1]$. To help visualize this, let's write them in an array. For instance, maybe they look like
$0.01101000...$
$0.10101100...$
$0.11111111...$
$0.01100001...$
I want to build a real number $s$ in $[0, 1]$ that I know isn't on this list. This means I have a bunch of requirements to meet: I need $s\in [0, 1]$, and I need $$s\not=r_1,\quad s\not=r_2, \quad s\not=r_3, \quad. . ..$$
The idea is that I'll define my real number so that each binary digit takes care of some requirement.
First, to make $s=[0, 1]$, let's begin with "$0.----$". There, that was easy. Now what about the other requirements?
I'm going to rewrite our array above, but with certain digits suggestively highlighted:
$0.{\color{red} 0}1101000...$
$0.1{\color{red} 0}101100...$
$0.11{\color{red} 1}11111...$
$0.011{\color{red} 0}0001...$
And remember that two real numbers are different if their binary expansions are ever different. (As remarked above, this isn't quite true on the nose, but ignore it for now; it's easy to fix after you get the general idea.) So e.g. to make sure $s\not=r_1$ I just need $s$ and $r_1$ to have one different digit, and so forth.
So here's how we do that:
Take all the red digits and put them together - this is the diagonal sequence. In this case it's {\color{red} $0010...$}. The $n$th number of the diagonal sequence is the $n$th digit of the $n$th real on our list (check this!)
Reverse them! Change each $0$ to a $1$, and each $1$ to a $0$. This is the antidiagonal sequence, and in our case is $1101...$
Put a "$0.$" in front of the antidiagonal sequence to get a real in $[0, 1]$ (here, "$0.1101...$"); this is our $s$.
Now clearly $s\in [0, 1]$ so it's enough to check that $s$ isn't on our list.
$s\not=r_1$, since $s$ and $r_1$ differ on the first binary digit: $s$ has a $1$ but $r_1$ has a $0$.
$s\not=r_2$, since $s$ and $r_2$ differ on the second binary digit: $s$ has a $1$ but $r_1$ has a $0$.
In general, $s\not=r_n$ for any $n$, since $s$ and $r_n$ will always have different $n$th binary digits.
So $s$ isn't on the list! And what we've shown, in fact, is that no list contains every real number.
(Except for that small detail about reals having two different binary expansions. You can try to fix this as an exercise on your own, or look at a full treatment of the proof in a textbook or online.)
Best Answer
The final result in the video is not wrong. But what much of the video says/does is wrong. $\infty !$ is a nonsensical symbol. Think of this as any other advertisement you see on TV: it may have some real things, but a lot of it is phrased misleadingly/deceptively to get people to click/view it.
The factorial is an arithmetic operation defined only on non-negative integers (as you have noted in your second sentence), and this definition is done recursively, i.e we define
So, we can only take factorials of non-negative integers $0,1,2,3,4,5,6,\dots$
Trying to somehow claim that $\infty !=\sqrt{2\pi}$ is then a nonsensical assertion based on the above definition (because $\infty !$ is not even defined based on our above definition).
The correct statement.
The correct statement (i.e no wishy-washy handwaving nonsense) is that Riemann's zeta function is such that $\zeta'(0)=-\frac{1}{2}\ln(2\pi)$. What does this mean? You say that the math used in the video is far beyond you, so fine, let me just tell you parts of "what" is being said.
Riemann was a famous mathematician, and one of his works was about a certain function. This function has the symbol $\zeta: \Bbb{C}\setminus \{1\}\to\Bbb{C}$, and it is what we now call "Riemann's Zeta function". What does this function do? Well, it takes a complex number $s\neq 1$ as an input, and it outputs a certain complex number $\zeta(s)$. The exact definition of $\zeta$ is not really important to us right now (and indeed giving a correct definition for this is something one can only do after a half-semester course in complex analysis). All you need to know is that one can prove that the function $\zeta$ is differentiable at $s=0$, and that \begin{align} \zeta'(0)=-\frac{1}{2}\ln(2\pi) \tag{$*$}. \end{align}
So, the correct statement is $(*)$. This is 100% correct mathematics (I didn't fully watch the video, so I won't comment on whether the Youtuber's explanation of this result is correct). But, rest assured that $(*)$ can be given a completely airtight proof. HOWEVER, what is utterly nonsense is going from $(*)$ to the statement that $\infty!=\sqrt{2\pi}$. This is what I was talking about in my first paragraph: there is some real mathematical content in this video (i.e the definition of $\zeta$ and the calculation of its derivative at $s=0$). However, the video is also full of "false advertisements" in that it makes absurd claims like $\infty!=\sqrt{2\pi}$.
One has to really turn a blind eye to several mathematically incorrect steps (such as calling $P=1\cdot 2\cdot 3\cdots$, which is really just $\infty$, and doing arithmetic with that), and then hand wave alot of details to even heuristically try to justify the relationship between the correct equation $(*)$ and the nonsensical $\infty!=\sqrt{2\pi}$.
Final Remarks.
The zeta function is also notorious in "everyday math" claims, because of the somewhat (in)famous $1+2+3+4+\cdots =-\frac{1}{12}$ nonsense. Anyway, to finish off, the statement that $\infty!=\sqrt{2\pi}$ makes as much sense as saying "the earth and moon are both round, therefore, the earth and moon are the same thing".