I think the terms "Final Four" and "Frozen Four" are NCAA-owned. "March Madness" used to be co-owned by the NCAA and the Illinois High School Association, who came up with the term originally for its high school basketball tournaments, but it's owned completely by the NCAA now, with the IHSA still allowed to use it for its tournaments.
"Sweet Sixteen" is owned by the Kentucky High School Athletic Assn., who licenses it to the NCAA. I haven't seen any trademark info for "Elite Eight."
Bowl games in football are owned either by Disney/ESPN (who owns a lot of the smaller bowls outright), or by the other individual bowl owners. For example, the Rose Bowl game & name are owned by the Pasadena Tournament of Roses Association. AFAIK, none of the bowls are owned directly by the NCAA.