It seemed like the one who came from the store was trying to pass as the “real” Buzz (Andy’s) to...try to foil their plan? Or something? But then this got dropped and they were all friends. What the heck?
shareTo add conflict. Plus for the humor of Andy's Buzz and the rest of the gang being annoyed with him.
shareOut of interest, did you see the first Toy Story?
In the first movie, when Andy first receives Buzz, Buzz doesn't realise that he is a toy. He believes himself to be an actual Space Ranger named Buzz Lightyear. (In Toy Story, the Buzz toy is meant to be based on a cartoon character.)
In the second film, when Andy's Buzz is in the toy shop, he meets another Buzz toy who is straight out of the box. So this new Buzz also believes he's the actual Buzz Lightyear just like Andy's Buzz did. He also assumes Andy's Buzz is an impostor, and thus must be somehow associated with Zurg. (Buzz's sworn enemy in the cartoon.) That's why he captured Andy's Buzz, it had got nothing to do with derailing the other toy's quest for Woody. He didn't know anything about that.