In terms of big game animals, I think what a person considers a "trophy" animal not only varies between each hunter, but changes as we gain hunting experience. For many of us, our first of any animal is a valued trophy to us when we kill it, however after 5 or 10 years of hunting those animals, we would probably hold out for a larger animal.
For example, my first mule deer was a 1 1/2 year old spike buck. I couldn't wait to get those antlers home and hang them on my bedroom wall. But now, after 45 years of deer hunting, I wouldn't consider shooting a spike buck, and the past few years I've passed on easy, makeable shots on many 5x5 whitetail bucks hoping I'd find one bigger.
Most of my hunting has been DIY. It wasn't until I hunted in Canada and Africa that I hunted with a guide or Professional Hunter. When you hunt with a guide or PH, they know the animals and they will tell you the trophy "quality" of the animals that you see.
After 3 trips to Africa, I've shot over 34 African animals and I've had a couple dozen of them mounted. And even though they all didn't qualify for SCI or Rowland Ward record books, because of the memories of those hunts, the beauty and uniqueness of those animals, they are trophies to me.