You’re skipping the idea that there is such a thing as the eigenfunction of the derivative.

Because taking the derivative is a linear function, a × e^x is its own derivative for any constant a.

That means, for example, that e^(x+1) or e^(x+π) are their own derivatives, too, as those are different ways to write e × e^x, respectively e^π × e^x.

e^x just is the ‘simplest’ of those functions (ignoring the most simple one: f(x) = 0, but that one isn’t representative for all the other ones)