If a spell or ability causes you to put a card into your hand without specifically using the word "draw," it's not a card drawn.

The triggered ability can trigger only once each turn. It doesn't matter if Mystic Skyfish was on the battlefield when the first card was drawn. If it's not on the battlefield when the second card is drawn, the ability can't trigger at all that turn. It won't trigger when the third or fourth card is drawn.