Not-Unconscious does not necessarily mean awake and aware.
Sometimes the game just expects us to understand certain terms without giving explicit rules text about what they mean. Famously, 5e doesn't specify a 'dead' condition or give any rules about what it means to be dead. So there is a precedent for the game having certain game concepts left to the players' understanding of the world without giving specific rules text.
The Unconscious condition has specific effects that don't jibe with petrification: Unconscious creatures fall prone, drop their belongings, and take automatic critical hits, none of which makes a lot of sense with a creature being turned to stone. So there's a good reason not to bring those into a petrification situation.
Which is a all a rather long way to say 'the DM decides that', but:
What rules text even represents being mentally shut down?
Other than the actual name of the condition, the "incapacitated, can't move or speak, and unaware of surroundings" bullet point is as close as the Unconscious condition ever gets to specifying that your mind is shut off, and that text is present in Petrified as well.
So, for whatever it's worth, it seems like Petrified includes the mental component of unconsciousness, in asso far as that state is represented in rules text at all.