From what I understand, there are different factors that influence a cats behavior towards smaller animals:
Hormones: If a (female) cat has given birth to a litter, her hormones are different than usual (just like in a human mother) to make her bond with her kittens. Especially the hormones for love and bonding (Oxytocin) are very high, which would allow a cat to bond with a bird or mouse as well.
Feeding: A well-fed cat has less incentive to hunt than a hungry cat. Most cats that were born in human care (especially indoor cats) don't even learn from their mothers that rodents and birds could be food. Some cats may have weaker hunting instincts and never start hunting small animals (others with stronger instincts certainly can start a habit of hunting, but most don't eat the prey after killing it if they haven't learned that as kittens).
Behavior of the prey: Hunting instincts are triggered by certain behaviors of the animal the cat observes. Those are mostly shuffling noises rodents make when moving around, high pitched noises rodents and birds make, and alternating sudden, jerky movements with remaining motionless. If an animal avoids those patterns (like a bird with a broken wing that freezes in fear), there's a chance that the cat doesn't perceive it as prey or a toy.
All things considered, cats befriending birds or mice is a very rare phenomenon, they are more likely to befriend dogs and rabbits (as demonstrated in this question). Inter species friendships seem to be more common in dogs and herbivores like bunnies, guinea pigs or goats.
Since I don't have personal experiences with a cat befriending a mouse or bird, I don't know if they still hunt other birds or mice.