When Keith Lee returned to action a few weeks back with the "Bearcat" nickname, there was a lot of talk about who came up with the nickname. Some assumed that Lee came up with the name but that was not the case at all.
Following the news of his release from WWE, Lee responded to a fan who claimed that the gimmick was Lee's idea. Lee quickly shot that down and said, "No. It was not. Lol."
The nickname was first used by Edward "Bearcat" Wright, a wrestler who was popular in the 50s and 60s. He was inducted posthumously into the Legacy wing of the WWE Hall Of Fame in 2017 so it's possible that someone in WWE remembered the name and decided to use it for Lee.
As for Lee's status, since he was a main roster talent, he's likely under a 90-day non-compete so he won't be free to sign with another wrestling company until February. Lee's wife Mia Yim was also among the 18 names released today.