Why is information not doubled in this context?
If I roll an eight-sided die, then I could represent the result with three bits.

If I roll two eight-sided dice, I have generated double the information -- and I need six bits (three bits for each die) to represent that information.

If I have just four bits, I could represent the result of rolling a 16-sided die, or the result of rolling an 8-sided die and flipping a coin -- but I could not represent the result of rolling two dice.

This is why information theory represents a basic unit of information as a bit -- because it fits with our intuitive understanding of what "X times more information" is.  Twice as many books is twice as much information (presumably), two phone conversations is twice as much information, etc.  Twice as many books, or twice as many phone conversations, takes twice as many bits to represent.  Hence information theorists equate one binary bit with one "bit" of information.