A trophy is a symbol of an accomplishment. Now, I'm not a competitive person, but I would say that people who compete do so for the sake of being recognized as the best. It is only on television that I have seen people obsess about a trophy qua trophy.
I have a book called The Simpsons and Philosophy, and all but two of the essays in it are very good and worth reading. (Those other two are boring drivel that robbed me of precious moments of my life that I could have wasted doing something else frivolous.) My favorite of the essays is "Enjoying the So-Called 'Iced Cream': Mr. Burns, Satan, and Happiness," by Daniel Barwick. To paraphrase, one of the reasons that Mr. Burns is unhappy is that he sees the world in abstractions. F'rinstance, in the episode where he joins Homer's bowling league, he does so only for the sake of acquiring the trophy, and completely misses half the point of bowling leagues: hanging out and drinking with your friends. When Mr. Burns takes the trophy, it becomes a symbol for an accomplishment he really had no part in. "[T]he result is that the original thing that is symbolized ceases to exist, at least in any pleasurable way. Unfortunately for Mr. Burns, it is the original thing that he truly needs for happiness."
This is not to say that this is how T-Rex is acting. If anything, I would say that it works in reverse: because the trophy symbolizes an accomplishment, by having the trophy, he can pretend to have accomplished something he didn't really do. Or, he might have an ironic perspective on it, thinking it rather funny that he has a trophy for winning a track meet.
Utahraptor steps in, though, and makes T-Rex realize how awesome it would be to instead making novelty trophies. The intention may have been to prevent T-Rex from filling his house with trophies for things he didn't do, or just plain ruining everything forever. The trophy then becomes a souvenir for a friendship, allowing it to once again symbolize something real and actual. Even if Dromiceiomimus can't really appreciate hers. And now that I think about it, I am pretty sure that when she dies, unless Utahraptor can stop him, T-rex will make her epitaph read, "DROMICEIOMIMUS: THE CLASSY DAME WITH THE SIX-SYLLABLE NAME". He'll go on explaining, "Yeah, that's totally what she wanted. She even had a trophy with that inscribed on it!"