The square root of $y$ is usually defined as the positive solution $x$ to $y=x^2$, so the negative variant is not considered.
In the same way, the inverse cosinus and sinus give the solution on respectively $[0,\pi]$ and $[-1/2\pi,1/2\pi]$, and not all solutions.
But the complex logarithm is often considered as multivalued, why?
Is there any mathematical or historical reason?
Strictly speaking, equation $x^2=y$ has two solutions, in both real and complex domain. Exceptional $y$ is $0$ when the equation has only one solution. The difference between the complex and real case arises when we want to choose a solution $x(y)$ to depend CONTINUOUSLY on $y$. In the real domain, if we restrict $y$ to some interval that does not contain zero, we have TWO continuous functions $x_1(y)$ and $x_2(y)$ that solve the equation $y^2=y$. In the complex domain, the same happens, if we restrict $y$ to certain regions which do not contain $0$. Such regions are called simply connected. In a simply connected region, equation $x^2=y$ also has two complex solutions.
However, in the real domain every region (connected open set) is simply connected, while in complex domain this is not so. This is the main difference. The situation with other inverse functions is exactly the same. For more details, see any undergraduate textbook which is called "Complex variables".