Is it allowed to define a number system where a number has more than 1 representation?

Sure, you can define anything you care to. In fact many commonly used systems, like decimal numbers, have elements with multiple representations like $0.999\ldots=1.$ Another example would be fractions, where $1/3=2/6.$

See A066352 in the OEIS for an example of an early use (by S. S. Pillai) of this particular system, and see also the related sequence A007924.

It is certainly allowed for a number to have multiple representations in a system of representations. In fact, the decimal system which we use all the time has this property. The well known fact that $$1=0.999...=0.\overline{9},$$ is an example of this. Whether there is any practical use to the system you mention I don't know, but it seems unlikely. In the decimal number system every number has at most two representations, and it is easy to see whether two of them represent the same number. In your number system the number of ways to represent a number grows very quickly as the numbers grow, making it a nontrivial exercise to even figure out whether two representations represent the same number!