The sports teams at the Catholic school in our town are called the "Eagles." Is that the kind of thing you mean?
For our school in particular, they used to be called the "Warhawks" but the diocese decided that wasn't really an appropriate name for a Catholic school, so they kept the same mascot symbol and changed the name to "Eagles."
Eagles are a sign of strength, courage, dignity, and are generally associated with power. They can also be seen in different religions or belief systems to represent different things.