I agree with the suggestion and let me add that the achievements could be like the Pokemon Black 2 and Pokemon White 2 Medals which gave you a box of medals and these were stored inside of them. as an achievement was completed they gave you a medal to show that you met said achievement
What also occurs to me is that to show the favorite medal to other players, a thumbnail of said medal could be placed at the top, next to the name, and when the mouse is placed on it (or the finger in the case of mobile device players) it will open a text that says the merit of the medal.
Here is an example of where the thumbnails of the medals that I say could go
The image was taken from a random match.
Another thing that occurs to me is that the medals could have extremely difficult or even unconventional achievements such as "capture a pokemon WITHOUT USING ANY ATTACKS", "Raise two shinys YOU CAUGHT", "Fight ALL LEADERS in the region *region name* WITHOUT USING FLY to change cities", "Briefly HAVE *Mewtwo/Rayquaza/Arceus/Keldeo* on your team at least once", or missions with that degree of difficulty or a little easier
Pd: I apologize if most of what I said was not understood, it is that I speak Spanish and I had to use Google translate to translate my comment 😅