The Elite Badge can be obtainable by buying a Subscription in KoGaMa. Buying a subscription grants you many perks, such as a Golden Frame on your Avatar Icon, Daily Gold, or XP Boost and even more! All Subscribers Perks are listed in the Subscription page.
You still keep badge when the Subscription expires.
There was an older version of the current Elite Badge. The badge was first planned to be named "Member", but its name has been changed to "Elite". You don't get a badge for the trial version of the elite (you used to get it)
This badge has been around since the beginning of 2019, a lot of people were buying it at the time.