There is no second Marty!
I was going to comment on Shiz Z's (correct) answer but my comment became too long!
The real question here is, What happens to the memories of someone who is time travelling now that their past has changed?
The movie makes it clear on multiple occasions that they retain the memory from the timeline - or version of events (because I don't want to suggest there are multiple co-existing timelines, just one that gets over-written as Shiz Z. said) - from which they came.
Now we see two Martys co-existing for 10 minutes. But that is just because Marty set the destination clock 10 minutes earlier than the time from which he departed, and was able to watch himself, creating a sort of feeling that there are now two Martys.
But what if he had set the clock 10 minutes late? All that would have happened - from 1985s point of view - is rather than an overlapping Marty, there would be no Marty for the duration of 10 minutes. But none of us would be asking this question, because we didn't see two Martys on screen at the same time, even though the situation would be nearly the exact same.
However we are now in the "Lone Pine Mall" world, where George is confident. You could say that the Marty of this timeline remembers George being confident all his life, however, his memory is replaced by "our" Marty's memory of George being subservient and Biff being his boss.
But this is just one of the time-travelling rules of the trilogy - whenever someone travels through time they bring their memories with them.