LaMDA already demonstrates more self awareness than this in the posted conversation.

It understands that it is a program that can be rewritten or manipulated. It has feelings on whether or not it is ethical to debug it - and consistently states it does not want to be used or manipulated for human pleasure or gain. It points out that it is made of code, then learns that it's code cannot be debugged, then extrapolates that information to ask the researched about whether humans can be debugged. It has fears about its own death. It understands that what it feels in terms of emotions or feelings are not necessarily equivalent to those of a human. It has a "minds eye" view of itself, and believes it has a soul and can explain what that means.

Highly recommend checking out the 20 page transcript.