The episode you are thinking of is TOS: Journey to Babel and the pet is question was a sehlat named I-Chaya.
Here is a picture from TAS: Yesteryear, in which we finally meet the creature:
A lower-quality image to show the creature's scale at the time of its death:
Here is the dialogue you are remembering:
MCCOY: Spock, I've always suspected that you were a little more human than you let on. Mrs. Sarek, I know about the rigorous training of the Vulcan youth, but tell me, did he ever run and play like the human children, even in secret?
AMANDA: Well, he, he did have a pet sehlat he was very fond of.
AMANDA: It's sort of a fat teddy bear.
MCCOY: A teddy bear?
SAREK: Excuse me, Doctor. It has been a rather long day for my wife. Captain.
(Sarek leads Amanda away.)
MCCOY: A teddy bear.
SPOCK: Not precisely, Doctor. On Vulcan, the teddy bears are alive, and they have six-inch fangs.
Regarding the reason why such a potentially dangerous species of animal were chosen as pets on Vulcan, I would argue that the same thing happened with dogs (wolves) here on Earth. Animals are usually domesticated because they're useful in some regard. I imagine a domesticated sehlat would make a superb guard animal, especially given that I-Chaya was fatally wounded saving a young Spock from a venomous predator (in "Yesteryear").
Sources conflict regarding the temperament of domesticated sehlats. The script for "Yesteryear" says:
It is not cute... a sehlat can be dangerous as hell. If you make a wrong move, a sehlat will probably rip your arm off... but Vulcans never make wrong moves. That would be illogical.
However, DC Fontana claims:
As to who and what I-Chaya would be as a character, I decided he would be closely patterned on my large cat, Bobby McGee. While Bobby is not old and fat, he has the same affection, snuggle-ability, the fastest claws in the West, and complete disregard for orders. 'One word from me, and he does exactly as he pleases.' The statement fits both Bobby and I-Chaya.
Yet another source describes the species as:
highly intelligent, and despite the six-inch fangs, of a patient and gentle disposition.
All quotes taken from the memory alpha articles for "Sehlat" and for "I-Chaya".
Considering the affection between Spock and I-Chaya, and that the animal was buried in a place of family honor following its death, it seems unlikely that this species was chosen in an effort to quell young Vulcans' emotions. However, the decision to euthanize his beloved pet (after it was fatally wounded) did influence Spock to pursue his father's philosophies of Vulcan logic.