The question is not how an artist from that culture would portray the creature using their traditional art idiom, but rather how the majority of people in that culture would actually visualize it in their mind's eye.
My friend included a series of images of eagles to frame the question. Everyone presumably sees an actual eagle with a similar set of biological eyes, but eyes are not like cameras, and the mind begins shaping the image even in the act of seeing.
What becomes of that person's mental image of the eagle when it is shaped by memory, imagination, stories, and the exposure to art? When he or she hears the Eagle character mentioned in a folktale, what form does that image take in their mind's eye?
If we start by looking at art, there's a lot of variation in the degree of abstraction, even among traditional cultures. The animals in the Lascaux caves have anatomy that is carefully observed and described, despite its simplification. Other art styles like Aboriginal Australian and Native American petroglyphs are very abstract.
Growing up in our times, our imaginations are influenced by a kaleidoscope of comics, animation, movies, illustration, and the Western obsession with realism.
A set of related questions emerges. Would a person who has never seen a photograph recognize the eagle in the photographic image more readily than they would in their culture's traditional representation? How does a person's imagination of their mythic beings change once they have been exposed to photography and other art styles?
Is abstraction or stylization more psychologically realistic? Even to our modern eyes, accustomed to photographic realism, the abstract image sometimes telegraphs its meaning more directly:
Finally, let's apply this thinking to a practical problem. Supposing there were a certain culture with a rich oral tradition but very little visual art and no exposure to worldwide art traditions. If a contemporary artist was given the assignment of painting a mural to represent their myths, how should he or she try to depict it?
(Example above: When John Singer Sargent was asked to paint the murals in Boston to visualize ancient stories, he could have painted them naturalistically, but instead, he drew upon the styles of Assyrian, Medieval, Egyptian, and Greek art.)
Should it be painted "realistically" according to the artist's own sensibilities, or should it be stylized to better represent how that culture might have imagined the creatures of that story?