Except that it is not.
A lot of different species can interbreed, and a lot of animals in the same species cannot.
There are a huge varieties of dog breeds, many cannot viably mate unless through intermediate dog breeds.
Loads of birds rely of feather patterns to keep them uninterested why being perfect genetic matches for interbreeding.
The preponderance of evidence suggests that humans could interbreed with other great apes. We have done it in the past, hundreds of thousands of years ago, and some scientists have stated that there is a decent chance that it would work (the Nazi's got increadibly close to trying).
And yet I think all of these categorizations are correct. Dogs are all quite related and since there is a range there is no anyway sensible cutting point. They have to be the same species or we need a new word to call it. The birds do not naturally interbreed, the is the whole point of the feathers. And just because many of the great apes can interbreed does not make them one giant species.
At the end of the day "species" is a layman term and is used loosely and with great flexibility.