A multivalued function is not a function. Hence the need to call it something else. In complex analysis you have branch cuts and things like that. We don’t call something a function that doesn’t have a unique output for a given input.

A set-valued function is a perfectly valid kind of function, and a multivalued function in complex analysis is just a set-valued function with certain continuity and other restrictions.

If set up that way then yes it meets the definition of a function. I’m not a complex analyst but I imagine they think of a function from C to C on a given branch rather than multi-function being a function from C to the countable product of C (or however one wants to formally define the set value). I think this because they tend to care about analytic properties and saying function from C to countable product of C includes far too many functions. Saying multifunction will immediately make one think of branch cuts.

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.

From the amount of complex analysis I've seen, I think I agree with you.

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.