In Ahlfors' Complex Analysis, chapter 3, section 4, the author claims that a region whose boundary consists of two circular arcs with common end points is either a "circular wedge" or its complement, he also claims that "its angle may be greater than $\pi$". Later on he refers to the complex plane with the interval $[-1,1]$ removed a "wedge with the angle $2\pi$"
My question is, what is exactly a circular wedge (or an non-circular wedge)? and how do you measure its angle?
A circular wedge is a domain whose boundary consists of two arcs of line/circles (the term that can mean either a circle or a line) with common endpoints $a$ and $b$, called vertices. The arcs are allowed to coincide.
The fractional linear transformation $z\mapsto \dfrac{z-a}{z-b}$ transforms the boundary into the union of two half-lines that go from $0$ to $\infty$. The domain itself becomes a sector of opening angle $\alpha\in (0,2\pi]$. Observe that the angle is the same at $0$ and at $\infty$. Since conformal maps preserve angles, the original wedge domain also had angle $\alpha$ at each vertex.
An example of conformal transformation of a circular wedge domain can be found in https://math.stackexchange.com/q/403075/79365.
I don't think there is a formal notion of a "non-circular wedge".