In the club search section why do some clubs have avatars showing next to the club’s shield icon?

Select your own avatar portrait by tapping on your portrait and then tap on the pencil next to the picture. Some avatars are free, some are bought in the Cafe Tips Shop, some are earned in club league prizes, and some are earned by completing the Memoirs Album. You can swap avatars that you own for free at any time. If you select to use your Facebook portrait instead, players that are Fb friends will see your Fb portrait, while non-Fb friends will see the character avatar you selected.

The answer to your question is: they selected an avatar from their list of avatar portraits.