It would have cost you nothing except a minute or so of your time to have refuted yourself before posting: https://arxiv.org/abs/1912.08274 https://arxiv.org/abs/1703.01700
(See also: https://en.wikipedia.org/wiki/Multivalued_function)
EDIT: Complex analysis was developed well before the formal definition of functions, relations, set theory. As such there are legacy terms in use. Mathematics has a lot of abuse of notation. For instance x=2 can mean assignment or an equation to solve. It depends on context.
In each branch of a multi-function the mapping is a function and, as a convenience, a single term “multi-function” is used to encapsulate all this. It cuts out the verbosity. A multi-function is not a function though.
My pedantry in this comes from the blog post. It was poorly written and left out some, from a mathematical point of view, important details. For instance the author uses a Schauder basis and has an uncountable sum at one point whilst ignoring the need distinguish between a Hamel basis and Schauder basis or the need to use a linear ordering in the reals to properly make sense of the uncountable sum.
They don’t think of Log(z) as a set of values. At least I don’t. I assume one means the principal branch when writing Log(z) unless specifically stated otherwise. Maybe in their mind they do think of an infinite set when seeing Log(z). I’m skeptical of this possibility though.
The insight behind multi-valued functions seems to be that for certain types of functions that seem unambiguous in the real case (roots, logarithms, ...) there's no clear candidate for which one is "the" logarithm function etc. So theorems look more like "let l be a logarithm function".
If you actually want to do complex analysis, it would not be very useful to have a vector valued function (even if that's technically possible) for logarithm etc. because most of the theory is developed for (in particular, differentiable) functions C -> C. So you'd rather just pick a branch.