A lot of people put a lot of significance on the moon. Scientists like to study the different moons. People have superstitions that are also based on the phases of the moon. Naming the moons would make it easier to determine which moon people are talking about. It makes communication easier.
Full moons aren't like hurricanes. They look cool but they aren't that significant. A blue moon or a lunar eclipse, maybe, but I don't see why we need to give the moons new names every month. It could probably help someone sell something, but other than that I don't see the need.
Since there is usually one full moon per calendar month, I do not want to need to know what each one is called. If we start naming each full moon, how long will it be before someone starts complaining about sexist or racist preference to the names? I don't think the world needs any more lightning rods.
Under no circumstances should full moons be given names. This latest trend of naming meteorological or other natural events is really getting very silly. On what level doesn't make one bit of sense to give a full moon I name? Seems to me this is more about people trying to get there 15 minutes of fame rather than letting nature do its thing without naming every single event.
Why would we start naming full moons? This happens at least once every month. It's nothing new. It's nothing extraordinary. They are just full moons, something we've experienced all our lives. If we started naming them, who would even care or keep track? What makes any one full moon different than the rest? I think it's a waste of time.