Amiga games with animal buddies
 

Chuck Rock 1 (flying reptile) & 2 (ostrich)
Bubble & Squeak
Fire & Ice (puppy shooting ice balls)
Wild Streets (wild cat)
Shadow Dancer (white dog)

any more?

Apidya: up to 2 small bees as options (Gradius-style)

Do the facehuggers (aliens) in Turrican games count? They are a kind of parasitic buddies ;)

They are not well behaved, just as frog-monster from Altered Beast which jump and cover your head.

Doofus - A dog

thanks for all suggestions

Yo! Joe! (giant bird)

Golden Axe (riding the fire breathing dragons)

yes and these not-dragons with beaks and strong tail

Sword of Sodan (giant ostrich-thing)
Neverending Story 2 (horse)
Wrath of the Demon (horse)
Back to the Future 3 (horse)

Dragon breed (you ride a dragon)

Shadow of the Beast 3 (big bird)
Shadow of the Beast 2 (water dragon)
Neverending Story 2 (furry-dragon-dog)

Defender of the Crown - Horse.
Superfrog - small green buddy.

Do those creatures from Cpt. Planet and the Planeteers count?
Also Dune - you can summon Shai Hulud and ride him.

Space Harrier bonus stage?
Does Stix in Bubba and Stix count?

Gods, the familiar, a flying bird.