The idea is that you invent a system where each integer corresponds to a finite bit-string, but the lengths of these strings must be unbounded. Unary is one such code.

Then you could set up a computer program which decodes these strings, so you feed it one bit at a time, each time asking it 'has an integer been uniquely specified yet?' The OP's claim is that whatever encoding method you come up with, he can come up with an infinite string that will make your program keep saying "no integer has been uniquely defined yet" forever.

So, nobody is encoding integers as infinite strings, although there's no particular why you can't, in fact the overwhelming majority of possible encodings use infinite strings.