I was watching the match again and noticed how the stadium (with the score 1-1) began to cheer when the referee announced that 6 minutes had been added. It seems that everyone was sure that Real Madrid could score one more goal, and in the end they did. As for extra time, everyone knew that Real Madrid would win, I'm sure even the City players understood this.
Not a few spectators who had left the stadium then ran to hear the equalizing goal and immediately it was like you said. Madrid fans believe that when they get a moment of added time, it means Madrid will win. It's extraordinary, watching the match like a world final and it was almost felt by us who could only watch it from the television screen. Feeling the immense power that was building within the Real Madrid players was something terrible for Manchester City at that time.
By the way, I saw the news that those fans who left the stadium at the end of the match were not allowed back even for stoppage time. I don’t know if this was done for ideological reasons or for security reasons (such abrupt returns of the crowd in the past led to cases of crush and death of fans), but whoever missed extra time and Real’s victory in this way will probably regret it all his life.