Well, unless his face is dirty with seeds/vomit, then no, he is not sick. It seems he has fallen in love with his own reflection and is trying to feed it, which for budgies, symbolizes love.
If he is lethargic, puffed up, and his face is dirty, then he is sick and you should take him to a vet right away.
However, based on what you described, it seems like he is exhibiting normal, if a little strange, budgie behavior. To discourage this, I would take out the mirror because he could get very aggressive towards you if he feels that he and his "mate" are threatened. Hope this helps!