In Jira the assignee field shows the "Fullname" of a user. When you map that field to Servicerocket you will get the username instead. It would be good if that could be Full name as it's presented in Jira.
See also SRSUP-6367
The current implementation uses the username, after reviewing this, we probably won't extend the feature to allow admins to choose the full name instead. However if there is more interest we may consider it in future.