When our team started we opted more to tie in with our town/community of Windham, New Hampshire which is how we came to be The Windham Windup. We knew we didn't want to directly use the school's jaguar mascot in our name because of how many teams throw "robo" or some other tech word before their mascot. It wouldn't be unique to us which is why we tied in elsewhere for the name but included the jaguar in our logo.
We also didn't use the school's colors of navy & gold but opted for close relatives in the color wheel of black & yellow. No one particularly liked the school's colors and we knew we wanted to wear polo shirts for a more professional look however the shop that did our shirts also serviced the school so we would have had the same shirts all the athletic coaches wore which wouldn't have gone over well.
I personally think schools care more about colors and care less about the name. It has been mentioned to us but our colors are so close its not a big issue.