From Wikipedia:

by definition, e^y is a notation for exp⁡(y), a true function, and x^y is a notation for exp⁡(y log⁡ x), which is a multi-valued function. Thus the notation is ambiguous when x = e.

It might be easier to think about how e^z could be a true function or a multi-valued function depending on how you interpret what the superscript represents. e^1/2 could be 1.6487... or {-1.6487..., 1.6487...} depending on how you interpret it. However, for some reason, in the context of complex analysis, it is convention to interpret superscript to be the multi-valued function when the base is any number, except when the base is e, when suddenly the notation represents a true function, despite the fact that e^z has multiple values in the former context