Summon Spectral Hunter

What happen if a Fate 3, skill 3 character with this card attached and it has placed 3 success tokens on it?

The destroy effect is a conditional passive effect.

The Fated rule seems to also be worded as a conditional passive effect.

Fated rule: When a card with the Fated X keyword has X or more success tokens on it, place that card on the bottom of its owner’s deck.

From the FAQ:

When card effects, passive abilities, or forced responses would resolve simultaneously, all cards that are affected resolve in the order determined by the active player, one at a time.

So in the asked situation, both passive's conditional's are met simultaneously when the 3rd success token is added, so the active player chooses which applies first, ie whether the card is destroyed or placed on the bottom of the draw pile.

Are the success tokens universal, or would each effect have its own pile of success tokens?

Success tokens are success tokens.

You only get separate piles if you they are on separate cards.

Had they wanted separate piles, this card could have been designed to stacked the tokens on itself, and then the interaction with Fated characters would have been different, but I think (given the expansion this came in) that getting rid of Fated cards faster was an intended effect.