The CW is attempting to rebrand the Arrowverse, and their new name is absolutely absurd. Arrow was originally intended as a standalone TV series, but in 2014 it launched its first spinoff. Fans initially alternated between calling this the "Flarrowverse" and the "Arrowverse," but they soon settled upon the second term. Soon, the producers themselves were referring to the Arrowverse internally (although initially they appear to have hyphenated it, i.e. "Arrow-verse").
Sadly, all good things must come to an end. The final season of Arrow has ended, with Oliver Queen sacrificing himself to reboot the universe. Sadly, as time passes it's becoming clear The CW is in no hurry to sign off on spinoff Green Arrow and the Canaries, which would have starred Katherine McNamara as Oliver's older daughter Mia; in fat, McNamara is reportedly in talks for the Urban Legend reboot, suggesting she's moving on. All that means there won't be any more Emerald Archers in the Arrowverse, and that name will soon become a historical curiosity. Consequently, The CW is attempting to rename the franchise.
Not very well, unfortunately. A recent promotional trailer showed off their various superhero TV series, celebrating them as "The CWverse." There's a sense in which this name is a little amusing, given only a few years ago The CW was wary of committing to too many DCTV shows at once lest they become seen as a superhero network. Unfortunately, it's also clumsy, and it certainly doesn't roll of the tongue. What's more, there's an odd sense in which it feels rather proprietary, as though The CW is declaring these shows as theirs. That sits uncomfortably with the fact both Supergirl and Stargirl started out on different networks and streaming services.
In truth, any concerted attempt to rename the Arrowverse is likely doomed to failure. The CW has failed to understand that this name wasn't really chosen by its producers, but rather was acknowledged by them. It was named by the fans, and there is no way the fans will accept the network imposing a new name upon it. Not least because, while The CW may be ready to move on from Oliver Queen, that doesn't mean the fans are. The best approach would be to simply leave it be, and wait until a few years have passed and the fan community itself begins to use another name. Alternatively, if there must be a rebrand, a better approach would be to cut across the board - to rebrand everything, from the DCEU to the Arrowverse. They can now all safely be advertised as part of the DC Multiverse, a brand that would embrace every DC movie and TV series ever released.
There is one other option, though, but again it requires a little patience from The CW. They could choose to brand their shows the "DC Universe." Granted, that name is currently used by WarnerMedia's streaming service, but that's pretty clearly winding down. It's just a matter of time before the name is free - and that would be a cool and useful successor to the "Arrowverse" brand, which would easily add or remove other shows and movies whenever the creatives please.