And honestly, while they don't always have to get along perfectly about everything I think the Batclan and Superman Family should know each other well enough that everyone should know each other's identities.
Clark knows the secret identity of Bruce's kids and I have no doubt that Bruce did his research on Clark's. Kara, Kon, Tim, Barbara, Dick know everything about each other. Dick has history with Clark, Kara is unofficially one of Bruce's kids, Tim is a detective and he and Kon are best friends and there's no secrets between them, Barbara is Oracle so she knows everything.
Cass knows about Kon because she read his movements and he told her but no one outside the Batfam knows about her, PG probably knows because she was Helena Wayne's BFF but I am not sure if the Batfam knows her story, Chris Kent knows Tim Drake is Robin and vice versa and he probably knows about Bruce.
I don't know about Huntress, Azrael, Jason, Damian, Natasha, Flamebird/Thara, Steel and Mon-El.
As for the non-crime fighters, Alfred knows (he's Alfred) the Kents don't know anything about the Bat family, and Lois knowing Bruce's identity is kind of ambiguous (although from Hush onward it is kind of obvious she does and Bruce knows she does too but they just humor each other). She used to be in the dark about it, during the whole President Luthor thing Lois and Batman sneak into the white House to get Lex's K-ring and Batman says something that leaves Lois confused and then is surprised that Clark hadn't told her his secret identity.
Also I am pretty sure Clark doesn't give his co-workers' identities to Lois so it doesn't surprised me that she had to figured out Babs = Oracle all by herself (or that she could, she's Lois freaking Lane after all).