You can define them to mean the same thing, but semantically they're very different. exp(z) means \sum_{n=0}^∞ z^n / n!, e^n means "e multiplied by itself n times". The former makes sense in any Banach space, e.g. matrices; the latter makes sense when n ∈ ℕ, and has a natural extension to n ∈ ℤ and a slightly subtle extension to n ∈ ℚ. There's then a unique continuous extension to n ∈ ℝ (by Kan extension)—but that's a formal continuation. It already strains the imagination to say "e multiplied by itself √2 times"; saying "√-1" times, or even "[matrix] times" is total nonsense. It's similar to how the Riemann ζ function is defined as ζ(s) = \sum n^{-s} when s > 0 and then analytically continued to s ∈ ℂ, but plugging in a general s to that sum is nonsense. *cough* Numberphile *cough*
Basically exp is important, and the homomorphism property trivially implies that exp(n) = exp(1)^n for n ∈ ℕ; this is the only reason that e = exp(1) is interesting. You can define exp as a power series (valid in any Banach space) and then define e = exp(1 ∈ ℝ); you can't define e first and then get the fully general version of exp from that.
