Wouldn't that be a big blow to the evolution theory? Two cannot represent the same thing ( kingdom classification ).
Well, evolution theory was never really in question, that birds are descended from dinosaurs has been common knowledge for some time now. It’s more taxonomy theory i.e. that birds are no longer considered a separate kingdom, but a class of indistinct weight, a class that is inferior to and part of another class called dinosaurs.
Optimally, you would classify animals based entirely on genetic similarity to each other, but seeing as we‘re unable to retrieve a complete DNA sample even from lifeforms that went extinct very recently (like mammoths), we may never know whether birds are genetically separate enough from dinosaurs to be considered their own distinct kingdom.
(I hope I understood your question correctly.)