A small piece of metal or plastic with words or a picture on it, that you carry with you or fasten to your clothing or something else, in order to show who you are, that you are a member of a group, etc. (Cambridge Dictionary)
A flat metal or plastic object, usually round, with a picture or message on it, that can be fastened to clothes with a pin, often in order to show that you like or support a particular group, idea, belief, etc. (Cambridge Dictionary)
A badge is a piece of metal or cloth which you wear to show that you belong to an organisation or support a cause. American English usually uses button to refer to a small round metal badge. (Collins Dictionary)
A device or token especially of membership in a society or group. (Merriam – Webster)
A characteristic mark. (Merriam – Webster)
An emblem awarded for a particular accomplishment. (Merriam – Webster)
A small piece of metal, plastic, or cloth bearing a design or words, typically worn to identify a person or to indicate membership of an organisation or support for a cause. (Oxford Dictionaries)
Examples
Round Badges

https://camaloon.co.uk/personalised-badges/round-badges
Asset Recovery Agent Badge

https://www.agentgearusa.com/product/asset-recovery-agent-badge/