(Upvoting and) seconding @Matt E's answer, in parsing texts and other literature, local holomorphy and global holomorphy are often insufficiently distinguished, thus understandably leading to confusions.(Years ago, it took me a while to understand (a) that my confusion was reasonable (b) how to resolve it.) Locally, except at $0$, there are two holomorphic square roots. Globally, which itself asks "on what open set? ... perhaps on what Riemann surface (complex manifold!?!)?"... the first (and archetypical) point is that, indeed, there is no square root of "z" on the complex plane/line. Ok. But, again archetypically/cliched-ly, on any simply-connected open subset of the complex plane/line not containing $0$, there is a (global) square root.
When one launches oneself into "Riemann surfaces", there is already some cognitive dissonance, reasonably-enough. The first point is that a given algebraic relation/function "$f(z,w)=0$" defines a finite-degree covering of "the Riemann sphere". The critical point for the question is that this can achieve the effect that a "function" only locally definable/holomorphic on $\mathbb C$ can become globally definable. Indeed, the cognitive troubles are amplified by the idea/fact that a given not-globally-definable function "defines" a Riemann surface... (This cracks me up... or not, given the many hours I labored to parse this cryptic mythology. :)
Eventually, one may discover that a "global" definition of a "function", e.g. defined by ODEs or by algebraic equations, that has "problems" about being pieced together globally, as in "covering space theory", "admit" a covering of the usual complex plane on which the "multi-valued-ness" pseudo-problems go away...
In summary: the traditional descriptions of the situations are pretty wacky, in my opinion!!!!! But, in fact, especially from our current viewpoint, it's not so crazy.