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.