The science community tries very hard, as described in my post about Birdwing butterflies and their food plants, to make sure each species is uniquely identified by a single Latin name. So long as the Latin names are correct, however, we can have as many local vernacular names as we like without causing significant confusion.
One species, one Latin name, many common names
One species may have several common names, even in the same language in the same region, let alone different languages and different countries.
A smallish ground-feeding black-and-white bird, Grallina cyanoleuca, is a good local example, being known as a Magpie-lark, Pee-wee, Peewit or Mudlark. None of those names are wrong, and any doubts about the identity of the bird can be resolved by pointing to one or referring to the Latin name.
Many species, many Latin names, one common name
Equally, one common name may be applied to two or more (usually similar) species.
The reason may be that the species are so similar that most people don’t notice any difference. St Andrew’s Cross Spider, for instance, is the common name for at least two species, Argiope keyserlingi and A. aetherea, which are so similar that even experts have difficulty. Two more, A. picta and A. trifasciata, are often called by the same name, too. Argiope kochi would be as well, except that it’s quite rare. They all look very similar and they all make the distinctive cross of white silk in the centre of their webs which gives them that name, so why not?
Or it may be that migrants and visitors applied an old and familiar name to an unrelated but similar species in their new country, e.g., think how many magpies there are around the world and how few of them are related to each other, or of all the “oaks” in Australia which are not oaks at all.
Or it may be that the name is descriptive and obvious enough that people came up with it independently. A brown snake has become a Brown Snake many, many, times, as iNaturalist shows.
Why and where do common names matter?
The whole point of common names is that they are the names known and used by ordinary people. As such, they are deeply embedded in local culture and history.
Just as importantly, they are the entry point for these people to further knowledge: people will look up “magpie”, not a new name invented by an expert group, to learn more about their local bird, because they won’t know the new name.
New ‘common’ names?
If iNaturalist, for example, becomes aware that two completely different species have the same common name, the organisation may need to separate them by changing one or both. A completely invented ‘common’ name is problematic, however, since it can’t serve the essential function of common names, i.e., being meaningful to ordinary people.
In that case, adding an adjective may be the best solution. It is, after all, what we do in everyday life. For example, when we’re talking about a certain lizard we may simply call it a Gecko; when that isn’t specific enough, we will call it a House Gecko; and when that isn’t specific enough, it’s an Asian House Gecko or an Australian House Gecko.
So the organisation can add a location to its common names by calling a pair of dragonflies the Australian Sapphire Flutterer and Asian Sapphire Flutterer (as happened after this discussion), or they can add any other pair of adjectives: Greater/Lesser, Black/White, etc, and anyone who knows the local common name can see immediately what’s going on.
The alternative is what the bird people do, insisting that common names in English should be unique, so that (e.g.) our Jabiru had to be re-named. There was already a South American Jabiru, Jabiru mycteria, so ours, Ephippiorhynchus asiaticus, is now a Black-necked Stork. That is informative enough but the change comes at the cost of all the cultural associations the old name had.