Suppose $f,h$ are continuous real functions on the unit circle where $f(-x)=-f(x)$ and $g(-x)=-g(x)$ for all $x$. What does it mean for $f$ and $g$ to have a common zero on the unit circle? Does it mean that there exists a $p\in S^1$ where $f(p)=0=g(0)$?
If so, can we ever discuss about a "common zero" if the co-domain doesn't include the point $(0,0)$? In other words, is the term common zero irrelevant for maps with co-domains of, for example, $S^1$?
A common $0$ would typically mean a common root, so you'd want $f(p)=g(p)=0.$ If $f,g$ have co-domains in $\mathbb{R}^2,$ and $(0,0)$ is not in the co-domain of one (or both) of the functions, then we cannot say they have a common zero. In short, both functions need to have a zero in order for them to have a common zero.
This doesn't mean the term common zero is irrelevant though. Say if $g$ doesn't have a zero, and we can show that some assumption implies that $f,g$ have a common zero, then we know that assumption is false. Proof by contradiction is a powerful tool.
