I believe you're referring to the medals that a user can get for completing a tournament in one of the top 3 positions. The small number is how many times they have gotten that particular trophy.
Does this answer your question or did what you see look differently? 
I'm not sure if those are the medals I'm referring to. I have played two games against a member named dhel975. In looking over her previous games, I saw a grayed-out icon that looks like a medal. Since it is grayed out, I cannot tell if it is the same as the ones you are referring to which are obviously very specific. There are no small numbers and virtually every game has one of these icons. I am wondering if this simply means that these games are tournament games and thus eligible for the kind of medals you are referring to. Thank you for answering.
I see that some games in my opponents' archives have an icon that is apparently a medal in their box. What does that icon signify? I cannot find anything about it on the Help pages.
Thanking You in Advance,
I Remain Very Sincerely Yours,
IGP1200