Does WWE own CM Punk's name?
Phil Brooks went by the name "CM Punk" years before he officially signed with WWE. Brooks popularized the name in the world of wrestling, and he managed to use it throughout his WWE tenure.
As CM Punk was the name he used from the late 1990s, WWE never had trademark ownership over the name. Looking at Justia Trademarks, Phil Brooks was the one who got the name "CM Punk" trademarked. He also renewed ownership in 2019.
The Straight Edge Superstar confirmed in a tweet that Vince McMahon and WWE never owned his name:
The possible reason WWE let CM Punk keep his name is that he signed with them in 2005 -- an era where WWE was liberal with Superstars using their real names. John Cena and Randy Orton are perhaps the biggest examples of this, with superstars like Shelton Benjamin allowed to use their real names as well.
It was quite different back then compared to now. In modern WWE, the company has a complete emphasis on trademarking names, especially those of Superstars. However, they do seem to look at it on a case-by-case basis.
AJ Styles, for example, trademarked his name all the way back in 2004 and has continued to use it in WWE.
CM Punk returned to wrestling after 7.5 years with AEW
CM Punk has been Phil Brooks' brand name for over two decades now. The AEW star was always firm about what he wanted, and although things didn't always go his way in WWE, he walked out with his head held high.
His absence from the wrestling ring is perhaps what made his AEW debut special. Thankfully, he will continue to use his trademark CM Punk name as well as the famous Cult Of Personality song by Living Color.