The first part of the sentence (Patterned after a cartoon character designed by Walt Disney himself) is a modifier. The first noun that follows is what it's supposed to modify. What this sentence seems to be saying is that the university was patterned after a cartoon character. It's the dreaded dangling modifier.
Am I being too Mickey Mouse about this?
Dangling modifiers are a problem because they are illogical. At best, they make sentences hard to read. At worst, they sound really silly.
To fix a sentence with a dangling modifier, you generally need to rewrite it. For example, the following would get rid of the dangling modifier:
The University was granted permission to use the Duck, which was patterned after a cartoon character designed by Walt Disney, as a mascot many years ago.
But it doesn't read very well. Major surgery often helps:
The University patterned its mascot after a cartoon character designed by Walt Disney himself. The school received permission to use it many years ago.