I first thought that Zelda was Link until I played A Link to the Past which was my first Zelda game ever. I didn't know the Hero's name was actually Link until I played the Adventure of Link some years later and learned that was his name.
No, I knew from the start that Zelda was the princess and Link was the hero. I can see how people might think Link's name is Zelda, that being the name of the series and all, but it's mentioned at least once in almost every Zelda game that the princess is named Zelda, and it wouldn't make much sense to have two characters named Zelda. On the back of the game boxes, it usually says something like, "Join Link on his most epic adventure yet!" or, "Play as Link while you travel across Hyrule." They don't put the name of the main character there for no reason whatsoever!
And I'm going to restrain myself from ranting about how some people still can't get his name right, even when they are informed that his name is Link.
I thought his name was Zelda when I had only seen my brother's friend play Ocarina of Time once and had seen his character in the original Super Smash Bros. I was confused why it kept saying Link until I realized that his name must be Link. Oh the joys of childhood.
Funny story actually, my older cousin made me a file on ALttP and named it after me of course. So for awhile I thought he had the same name as me, then I started a new file and realized you could name Link lol.