It's a poor analogy. A really poor analogy. When people say that, the idea they are trying to get across is that there is a limited total supply available to acquire and that initially access to that supply is eaiser. As time goes on, the easy to access supply is all acquired, and the remaining supply becomes progressively more and more difficult to get.
As far as I know, double hashing does not protect against birthday attacks. Where did you hear that?
The protection against birthday attacks is the incredibly large number number of possibilities in the address set, and a (hopefully) even distribution of addresses across all 160 bit values.