The Justice League and Batgirl both have J.K. Simmons as Commissioner Gordon. It's strange, but JK Simmons always seems to experience this. In fact, he explained how strange it was that he reprised a character for both Marvel and DC. To quote his interview with Den of Geek:
"I was as surprised as I was when [Marvel] came back to ask me to do [Jameson] that they were coming to me to play Batgirl's father, Commissioner Gordon again, in this new version of the universe. We've already had lots of discussions about the script and how to play everything."
Furthermore, he answered whether Batgirl and Justice League share the same universe.
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Does Batgirl share the same universe as Justice League?
Batgirl and Justice League do not share a universe, but it sounds like DC recast him for a reason. A multiverse merging reason.
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Since Batman V Superman, DC has been playing with the idea of its multiverse. In the interview, he discussed the multiverse further as follows:
"[I'm] Looking forward to diving back into the DC multiverse... When we were doing Zack Snyder's Justice League, that was with the intent that we were laying groundwork for these characters to be expanded on. And with these superhero multiverses, I guess now they go in different directions."
Has J.K. Simmons become the staple of the universes? No. However, DC may use him to keep some things the same in the multiverse.