This is a naming scheme that has been used for moons since they were discovered. For example, since their discovery all the way to the 20th century, Jupiter's moons were known simply as Jupiter I, Jupiter II, Jupiter III and Jupiter IV. As new moons were discovered, this practice was followed. Even for the four most easily observable moons of Jupiter, the naming scheme surived for about three hundred years; many of the others only got their name in 1970s or later.
Remember that before heliocentric (or geocentric) models came to a good description of the solar system, people didn't even know that planets had an ordering. They were simply "wanderers" - stars that moved in relation to the "fixed" stars, and in predictable (though sometimes complicated) patterns. So they were named long before the ordering was discovered, and they were often given divine qualities (which still survive today in e.g. Astrology). If planets were somehow only discovered in the 17th century, like the moons of Jupiter, it's quite possible we would have simply called them Sol I, II, III...
The main problem with this is that the ordering can change - or rather, we might discover a orbiting body between two previously discovered bodies. There are some workarounds around this, but either it means you need to change the names of already discovered satellites, or you preserve the order of discovery rather than the orbital order.
The formal naming scheme came from IAU in 1975. Since then, newly discovered satellites of Jupiter are supposed to be named after the lovers and favourites of Jupiter/Zeus. A 2004 definition expanded this to their descendants. Since the names no longer have any implicit ordering, you avoid the confusion of whether Jupiter III is the third moon, or the third moon that has been discovered, or the third largest moon, or having to name the moon between Jupiter III and IV something like Jupiter IIIb.
If you're exploring the universe fast with your flashy new FTL drive, it makes sense to use placeholder names. A numbering scheme would work until you have a better name, which would usually follow an actual colonisation or mining operation. In much sci-fi, planets are known under multiple names - some are official designations, some are local names, others are well-known nicknames. So a planet known as Rigel IV might also be known as "Jerryworld" to its inhabitants, but they would still use the "official" name when communicating with outsiders. At least if you preserve the numbering to mean order, it's easier to maintain interstellar maps - presumably, in such advanced interstellar civilizations, the ordering would only change due to astronomical cataclysms, which are far rarer than human naming :) If you never heard of Earth, which helps you find it in your star map - the name "Earth", or "Sol III"?
As for the somewhat popular names like Terra or Luna, I expect they're meant to symbolise how small one planet is in the galaxy, much less the universe. Latin has long been used as the international language of science (even in medieval times), so it's not really a poor choice of one name all of Earth could agree on. Having 200 widely used names for one planet might be rather inconvenient for a galactic civilization.