As of v2.6.5 of the Git add-on, the Jira user card is available on the following screens:
Hover the mouse pointer on the name of the user. A small information box containing information such as email, zone & time and avatar for that user will appear.
Click Activity to view the latest activities performed by that user.
Click More > to open more options such as:
The Git User Identity should be configured for the specific user. Otherwise, the user card information will not be displayed for that user.