"She gave him the thumbs-up sign." and "She gave him a thumbs-up sign." What is the difference?

It's not a case of different usages being appropriate to different situations, it's just different ways of thinking about the thumbs-up.

Using a treats the sign as an ordinary gesture, in the same way that we say She gave him a wave/a smile.

Using the treats it as a gesture with specific meaning ('all is well' or 'I'm OK with that'), as in the Masonic handshake, the Nazi salute.