Even animals such as rats can pass cultural information from one generation to the next. We know this not because we set out to study rat intelligence, but because rats have long outsmarted people trying to poison and kill them.
This was demonstrated by an experiment designed to study how to poison rats more effectively. A controlled colony of rats was fed two kinds of food, which we can call food X and food Y. At first, the rats ate equal amounts of both. One day, food X was poisoned with lithium chloride. Although neither food was poisoned again after that single day, no rat in the colony ate food X. Even more interestingly, no rat ate food X many generations later.
This colony of rats had a cultural taboo that was passed from generation to generation. It made this colony different from other colonies of rats of the same species. As with human taboos, it persisted long after the reason for it had disappeared. But the taboo still made sense: would you want to be the first rat to try food X again and find out whether it was poisoned, after no one had eaten it for generations?
There is also considerable evidence of much more complex cultural traditions among animals, especially tool use among primates. I chose rats for my example to emphasise that cultural traditions are passed to future generations by animals other than primates.
More answers: Vegan Ethics FAQ.