It's all to do with your ship's sensors and the planetary classifications they has been programmed with. Once a planet's environment falls within the fairly strict parameters required to classify it as "Earth-like", that is what it is labelled as.
Your ship's sensors are not sentient enough to recognize that the "Earth-like" planet you are looking at when you look at Sol 3 is actually the "Earth" from which the phrase "Earth-like world" is derived. You're the most sentient thing aboard your spaceship, it's your job to figure out these things.
It is also time-proof. Don't forget, according to current theories, for most of Earth's history, Earth itself was not "Earth-like": too hot, too cold, too much oxygen, too much carbon dioxide... and there is no guarantee that Earth will remain "Earth-like" forever, especially if one considers the possibility of hostile Alien incursion. It may be in FD's plans to take Earth and sterilize it, rendering it no longer "Earth-like".
Finally, of course, it would open the floodgates. Imperials would insist that their sensor suites report Capitol as "Capitol" rather than just another "Earth-like". The same with Turner's World (Alioth) and all the other popular capital planets. Soon the phrase "Earth-like" would lose it's usefulness as a definition as there would be so many special cases.