No, as username90 says, e was not created to perform rotations. The aspect of e that comes into play for the relation to sine and cosine is that the function e^x satisfies the differential equation df/dx = f. This makes the Taylor series convenient by not introducing factors of (ln K)^n in each term.

Thus, 2^ix = cis (x ln 2) = cos (x ln 2) + i sin (x ln 2). And since e is, by definition, the number that satisfies ln e = 1, e^ix can be stated more simply as cis x. And that identity, ln e = 1, is in fact the original motivating definition of e.