No he didn't? Turing's machines simply specified a language of symbols. It doesn't require the numerals 0 and 1, if that's what you're asking.
No we haven't? Once you start building electronic computers, the symbols 0 and 1 really lend themselves to an important electrical concept: On or Off. Binary is ubiquitous mostly because everything is modeled in a circuit and it's either On or Off right now and we use 0 and 1 to represent that state.