I did some manual testing to check what the different Canari Plush bonuses are. Let me know if any of these differ from your observations in game.

Red Canari Plush

These boost Exp. Points.

1. Level 1: 5% more Exp.
2. Level 2: 10% more Exp.
3. Level 3: 15% more Exp.

Gold Canari Plush

These boost the conversion of Prize Medals to Pokedollars and direct earnings from trainer battles.

1. Level 1: 15% more money.
2. Level 2: 30% more money.
3. Level 3: 50% more money.

Pink Canari Plush

These increase the Mega Shards you find.

1. Level 0: Small crystals drop 1-3, and large crystals drop 6-10.
2. Level 1: Small crystals drop 2-4, and large crystals drop 8-12.
3. Level 2: Small crystals drop 3-5, and large crystals drop 10-14.
4. Level 3: Small crystals drop 5-7, and large crystals drop 14-18.

Green Canari Plush

These increase your player's health.

1. Level 0: Your player starts with 100 HP.
2. Level 1: Your player has 150 HP.
3. Level 2: Your player has 200 HP.
4. Level 3: Your player has 250 HP.

Blue Canari Plush

These increase the modified catch rate prior to shake checks.

1. Level 1: Multiplier is 1.1x.
2. Level 2: Multiplier is 1.2x.
3. Level 3: Multiplier is 1.35x.

After you collect all Colorful Screws, you get a giant Canari Plush sent to your room. This appears to be purely cosmetic.