Don't only humans have the ability to pass on cultural information from one generation to the next?
Even animals like rats can pass cultural information from one generation to the next. We know this not because we were necessarily trying to study rat intelligence, but because rats have long been out smarting the humans who have been trying to poison and kill them.

This is illustrated by the following experiment which was designed to study how to poison rats more successfully. A colony of rats in a controlled environment was fed two types of food. Let us call these food X and food Y. Initially, the rats ate both types of food in equal amounts. One day, food X was poisoned with lithium chloride. Although neither food was ever poisoned after this one day, no rat in the colony ever ate food X again. What is especially interesting about this, though, is that no rat in the colony ever ate food X again even many generations later.

This colony of rats effectively had a cultural taboo which it was passing down from one generation to the next. This taboo made this colony of rats culturally distinct from other rat colonies of the same species. Just as with human taboos, this taboo persisted long after the reason for the taboo no longer existed. But it still made sense for this taboo to be maintained. After all, would you want to be the first rat to try food X again to see if it was poisoned, after no one had been eating it for many generations?

There is considerable evidence of far more complex cultural traditions in animals, particularly with regards to tool use among primates. However, I wanted to use rats in my example to emphasize that it is not just primates which pass cultural traditions to future generations.