Usually, I see $\aleph_0$ defined to be equal $|\omega|$. However, I don't really understand: in ZFC, everything should be a set. But $\aleph_0$ is not defined as a set, but rather as the image of a set under a function that does not appear to be well-defined for that set, since you define $|\omega|$ to be $\aleph_0$, but also $\aleph_0$ to be $|\omega|$.
Thus, what is the formal definition of $\aleph_0$ as a set?
Strictly speaking, $\aleph_0$ and $\omega$ are literally the same object, and cardinals are just initial ordinals. The "cardinality" map $\alpha\mapsto\vert\alpha\vert$ is just defined by sending each ordinal $\alpha$ to the (unique) smallest ordinal which is in bijection with $\alpha$.
The real purpose of the cardinal vs. ordinal notation is to handle notational ambiguity: $+$, $\times$, and $?^?$ are used for both the cardinal and ordinal versions of addition, multiplication, and exponentiation. Focusing on addition for example, this is why $\omega+\omega\not=\omega$ and $\aleph_0+\aleph_0=\aleph_0$ are both true; if we were being better about notation, we would do something like $(i)$ use "$\boxplus$" for cardinal addition and $(ii)$ write these two sentences as "$\omega+\omega\not=\omega$" and "$\omega\boxplus\omega=\omega$" respectively.
(Personally I really really hate the way the notation shook out, but what is one to do?)
Usually the definition is just that $\aleph_0$ is identically equal to the ordinal $\omega.$ More generally, for any set $x,$ $|x|$ is defined as the least ordinal $\alpha$ for which there is a bijection $\alpha\to x.$ (This gets more complicated when we don't have the axiom of choice and such an ordinal may not exist, but let's say we do.)
Moreover, a cardinal is just any ordinal that has no bijection to a lesser ordinal and it follows from the definitions above that for any $x$, $|x|$ is a cardinal, and that $|\omega|= \omega.$
As for why we use multiple notations to refer to the same object ($\aleph_0$ and $\omega$), the primary reason is just to emphasize the cardinal nature when it's being used in that context. Also, we can enumerate the infinite cardinals (which are a subclass of the ordinals, hence a well-ordered class) as $\aleph_\alpha$ for $\alpha\in \mathrm{Ord}.$ We can define this explicitly by transfinite recursion by the rules
$\aleph_0=\omega$
$\aleph_{\alpha+1}$ is the least ordinal that doesn't inject into $\aleph_\alpha$
for limit $\alpha,$ $\aleph_\alpha = \bigcup_{\beta < \alpha}\aleph_\beta$
and it's provable that the $\aleph_\alpha$ are all cardinals every cardinal is equal to one of them.
