We ( as in the network of alumni, students, faculty and staff) are two schools in Claremont, California who share Athletics and a Mascot [amongst other things].
In this digital age, search engine optimization (SEO) and search relevancy are increasingly important.
Searching for "Cecil the Sagehen" on any major search engine (and maybe even the minor ones!) results in a lot of matches where Pomona or Pitzer lay full claim to the mascot.
Maybe this was ok in the 1980s, but this doesn't work in the age of the internetz.
If both schools are sharing the mascot, both schools should share the search engine equity -- and both schools should note that Cecil is the mascot of their counterpart as well.
Each college can put their name first, but the sharing of the mascot really needs to be noted.