Could anybody tell me what it means when a cup/ trophy appears next to the name of a horse in the breeding barn

It's related to rewards. The horse was likely one of the top horses of the season when it retired or may be in your hall of fame. Being a top horse at the end of the season resulted in the trophy by the name in previous games anyway.

I've seen it pop up for a horse that wasn't either of those, though, so may be a little buggy or was at the time I noticed it.