"Relate" means to show or establish a logical connection between two things. When you relate A to B, either (1) there is already a logical connection between A and B, and you are making an effort to make it clear, or (2) you are establishing that logical connection.
I am writing a book that aims to relate the current political atmosphere to the country's history.
In this example, the book is meant to draw a connection between the two aspects: the current political atmosphere and the history, and it has an active role in the establishment of the connection.
In your sentence there is no need to establish a connection as the connection is there. Neither is there a need to show that connection. So "relate" is not appropriate here.
"Associate", on the other hand, means to think of things as occurring in tandem. When you think of A, B always comes to mind, then you associate A with B. It indicates a weaker and vaguer connection than "relate". It is not necessarily a logical connection but rather an unspecified and possibly coincidental co-appearance.
I always associate John with Mary, because every time I run into John, he is always with Mary.
John and Mary don't necessarily have any form of relationship. They just always happen to appear together when I see them, so I naturally associate them together. It is my impression.
Your passage at issue tells us that we have this idea that Aesop's fables are connected with childhood. It is our impression. But the thing that is responsible for having given us that impression is Victorians (Victorian writers/writers from the Victorian period). So, associate is more appropriate here.