His definition of Kolmogorov complexity is a bit loose. The rigorous definition uses Turing machines (or Minsky, or Post, or some sort of lambda expression, etc.) so the size is something specific. Different versions of complexity defined this way may give different values but have the same properties and asymptotics so one might just as well stick with the Turing kind. Chaitin's theorem (about the limit of Kolmogorov's complexity being just entropy) holds for all versions as well.

it's not just that they have the same asymptomatic. once you do the radix conversion (e.g. base 10 has log(10)/log(2) times more symbols), any 2 definitions are only off from each other by a constant