The Batcave in Arkham Origins is the real Batcave, as evidenced by Alfred being physically there in the first place and unquestionably cemented in the Cold, Cold Heart DLC.
Oh, why must I tell 'em again?
What must I tell 'em, my friend?
That we are born into sin
And we are worshiping murderers
Nothing but evil duplicitous men
I know you heard of 'em all
We should just murder 'em all
We should just murder and ball
We dance on the ruins we looting from all of they palace's walls