Hi Kaustubh, welcome on meta 
As far as I know, flairs appear only if the group is set as a “primary group”.
edit: it seems not as simple after testing. The “staff” default group doesn’t have a “primary group” option.
If you set an icon, it will appear next to the avatar as long as the user doesn’t belong to any “primary” group.
If they do, and the “primary group” setting is removed from this second group, the staff flair doesn’t appear instead, which seems weird (buggy?). It might have been raised before…
Here on meta, we have a team group, which is not the Staff group, and is used for our lil’ discourse flair among other things.