Why couldnt Rowling choose Potter to be like his mother Lily? To be a master of the sciences he undertook. Why couldnt he choose Harry to be as skilled as Snape or for that matter Voldemort or Dumbeldore? Its a question that begs asking cuz had Harry not been "mediocre" as Snape often describes him; had he not been amateurish with spells, potions, transfiguration, apparition he also wouldnt have been loveable!
Its a very unique feeling of warmth that you can feel within yourself as you identify yourself with Potter through each of the installments of the series. In hindsight, I say its our fascination for underdogs. Its the reason we all loved Rocky; the reason why we hate to see the Aussies win all the time. The very same reason makes us love Harry as he rides his luck to backfire the Dark Lord's Avda Kedavra.
I cant speak for others but for me its a feeling that makes me happy enough to chant Expecto Patronum!