If we take the approach underlined by Peano axioms, we get:
$$a + (-a) = 0.$$
Likewise, we can get the same thing in Euclidean plane:
$$|AB| + (-|AB|) = 0,$$
where $A$ and $B$ are points, and $|AB|$ is the length of the line segment $AB$ (thus the "operation" $AB - AB$ has no meaning, as we have not ascribed any attributes to the line segment).
On the other hand,
$$\vec{AB} - \vec{AB} = \vec{AB} + \vec{BA} = \vec{0}.$$
Well, what is the difference between $0$ and $\vec 0$ ? I think that a concrete definition of both should clear my confusion up. I understand that $\vec 0$ cannot be used in the same context as $0$ (because the former arises from the definition of vector), but I still don't have in-depth understanding of the concept of $\vec 0$. In Euclidean plane both $0$ and $\vec 0$ are points, right? So what's the difference then?
We sometimes write $\vec{0}$ for the zero vector in a vector space to distinguish it from the $0$ inside the field we are working over. For example, you could write that $0\vec{v}=\vec{0}$.
Actually, $0$ can also be used in vector spaces to denote the $\vec{0}$, because algebraically it is the zero element of the underlying abelian group. This is less specific than the notation in the last paragraph though.
Well, I wouldn't want to say it that blithely, given the context I mentioned above. In the Euclidean plane, things look like ordered pairs $(a,b)$, and we might say that $\vec{0}=(0,0)$ where the $0$ comes from the underlying field (not the plane.) These ordered pairs are coordinate vectors yes, meaning that they specify a particular point in the plane.