If AEW was airing or charging to watch old WCW footage then yeah WWE has all the legal right to go after them since they own the rights to the footage.
In terms of the name itself, WWE doesn't own the right to it due to letting the trademark expiring in the early 2000s. Its how MLW was able to get the War Games trademark and some other company trying to get the Halloween Havoc trademark despite WWE owning the footage for those events.
Think Cody still has priority if WWE were to file for the trademark since his filing is still active till this summer despite the refusal.
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