The Millenium Falcon's introduction to the plot is HORRENDOUS

I'm talking about more Rey knowing about it somehow if it's considered a myth or whatever.

 

Everybody's had great points on this.

I just would like to add that we need to remember some of the motifs of Star Wars - namely the one that means 'sometimes, the greatest treasures come in the most unlikely places'.

The Falcon is a prime example of this, as is Anakin himself in TPM, Yoda in ESB (originally but also in general - a small ugly elf being one of the most powerful beings in the galaxy), the Ewoks, and as much as it pains me to say....Jar Jar Binks.

So the introduction of the Falcon actually meets a strong motif that SW has always maintained. Don't judge a book by its cover...yada yada yada yoda, yada yada yada

 

I have to say all things aside like why Rey didnt know the name of the ship. The way it was introduced I thought was one of the best re-intro's of the movie. The perfect way to bring it back in is by making of fun of it. Its been that way in every movie its appeared in. Its a shotty old ship that everyone complains about but no matter what it seems to never let them down in the end. It did the same in TFA. I really liked the way they brought it in.

As far as the name thing goes, the ship is a YT-1300. It is not a Millennium Falcon. That name was given to it like how someone names their boat in the real world so perhaps someone just lost track of the fact that it was THE Millennium Falcon and not just a run of the mill YT-1300. Prior to Bala-Tik informing the First Order the ship was indeed the Falcon the officer who reports to Kylo even says droid stole a "YT model freighter".