I think families should have pets and share the responsibility. Giving a pet solely to a child and telling them to take care of it by themselves is a sure way to bury an animal (not trying to be totally pessimistic here, just my experience), but if the household has an animal (or animals) and everyone has some part in caring and loving that creature, not only will the animal thrive but the family will too. It is a good way to start to learn responsibility and how to care for another creature, but a child can not and will not dedicate all of their time to raise and keep an animal, they will need help, but they should be taught and shown how to do it.
As a child and a teen I always wanted my own pet that was separate from the family, but my parents never allowed it. I bought my first pet (a snake) when I was living in the dorms in college (it was forbidden but I hid him in my closet) when I was 19. Now that I have my own child and multiple reptiles, I do understand why my parents wouldn't allow me to have one separate from everyone in the household. Even now I can get lazy in the care of my animals, but I have my own wonderful family (my boyfriend and my daughter) who help out with all of them and their needs (thank goodness)!