Wolfram defines Global Maxima as :
A global maximum, also known as an absolute maximum, the largest overall value of a set, function, etc., over its entire range.
As per the definition, I'm not sure if $\infty$ can be considered as the global maxima for functions like $x$, $x^2$, etc. What if we consider the extended real number set?
You've probably found that when you write infinity in interval notation, you are instructed to write for example $$ y\in [0,\infty) $$ Note how the high bound infinity is not included. This is because infinity is a boundary not a real number, $$\infty \not\in \mathbb{R}$$ that is a number can approach but not touch infinity, so it could not be that the maximum is infinity.