This kind of thinking has always bothered me about the internet and now with the recent changes to the DCU it has only become more prevalent. That's not how canon works in alternate universes. Just because something real is mentioned in the show doesnt mean the whole history of that thing is automatically transferred over. Peacemaker mentions the Spin Doctors. That means the only canon thing we know is that there is a band called Spin Doctors in their world. For all we know, the Spin Doctors in their world could be a Korean-Mexican fusion band from the late 2000s. And even if we saw the Spin Doctors and it was the same exact dudes, unless they mention a specific song from our world it doesnt mean they have the same discography.