WWE held ownership of Cody Rhodes’ name after he left the company. This is why he’s just “Cody” in AEW, and he was previously only called “Rhodes” when introduced as “Cody and Brandi Rhodes.” Brandi never went by her name in WWE because they called her Eden for some reason.
According to Heel By Nature, WWE has cancelled their trademark on Cody Rhodes’ name. That theoretically gives Cody an open door to call his attorney Michael Dockins and get another trademark going.
Cody tried to register his name on April 13th when the trademark expired. Then WWE came back a month later and attempted to revive the trademark while blaming COVID-19 for the delay. It appears they finally gave up that fight.