If you look at the works of Bruni, Biondo and Vasari it is pretty evident that the contemporaries of Renaissance Italy (at least the intellectuals and artists) were aware that they were in a period of rebirth. They would often categorize history into three phases, antiquity, a state of decline (fun fact this is actually where we get the idea of the middle ages from), and a rebirth. They would always refer to their age as a new golden age, Vasari even specifically refers to the age he lived in as a rebirth.

I'm not sure how aware your average Giorgio would be aware of this, but I think it would be safe they were aware they were in a new age considering the boom in art, architecture and wealth.

I can't comment on the rest of Europe, but I think someone else could take that up.