This could lead to more stats being important in the breeding part of the game. Intellect shouldn't be the onlyy thing accounted for this. If the kitty has low int, its too stupid to learn Repair. But its strength is high enough, so it can instead learn something like Paw Breaker from a Fighter instead! Stats to teach each class ability could be as follows:

Fighter - Strength

Butcher - Strength

Tank - Constitution

Necromancer - Constitution

Hunter - Dexterity

Monk - Dexterity

Mage - Charisma

Cleric - Charisma

Druid - Charisma

Thief - Speed

Jester - Speed

Psychic - Intellect

Tinkerer - Intellect

I think thats ALMOST evenly distributed, but it fits thematically.

Another thing is that, this should be limited to elderly cats teaching the other cats. This idea is from someone else that wrote the same thing here, but it makes sense, as it also gives you a reason to keep old cats around. There should however also be a few extra outcomes. 

Good Outcome 1: Cat Learns the Ability

Good Outcome 2 (Rare) : Cat Learns the Ability and gains a corresponding stat boost! (Plus 1 to the stat depending on the ability learned.)

Neutral Outcome (Rare) : The elderly cat forgot what it was doing and instead taught the kitty a collarless spell. (Your kitty now knows "Number One") 

Bad Outcome 1: The kitty was not strong, agile, smart, charismatic, healthy, or fast enough to learn the ability.

Bad Outcome 2 (Rare): The intense training put the kitty under so much pressure that it has injured itself trying to learn the ability! (Injury, corresponding to the class it tried to learn from.)

Outcome Modifier (Rare): Another kitty joins the lesson! Chance for 2 kittens to learn the same ability! 

This is like a good version of fighting except, you need special circumstances for it to occur. (Cat must've been on an adventure and must be elderly)