Answer:
The more you take away from me, the bigger I shall get. What am I?
I have a strong affection for belly buttons.
I'm not a bird, but I can fly through the sky. I'm not a river, but I'm full of water. What am I?
I always have to work with something in my eye. What am I?
I am a protector. I sit on a bridge. One person can see right through me, while others wonder what I hide. What am I?
Those with eyes bigger than their stomach will definitely leave the restaurant with one of them. It is named after a house pet as well.