It's a much more useful correspondence than just an isomorphism. Take your base-2 2-adic integer. Truncate it to whatever bitwidth you want. That's the two's complement representation of that number. Unsigned pops out as a special case, obviously.

I've mostly seen this used by cryptographers to prove results over arbitrary bitwidths, since you can prove it over the p-adics and deal with truncation/division separately.