User fields, such as assignee can be automatically mapped in JIRA and TFS / VisualStudio Team Services, provided that both JIRA and TFS / VSTS server are connected to the same Active Directory instance and therefore share their user account data.
For instructions how to connect JIRA to Active Directory please refer to https://confluence.atlassian.com/display/JIRA/Connecting+to+an+LDAP+Directory.
To enable automatic mapping for "Assignee" field please click the "Values" link for "Assignee" mapping and select Mapping Mode "Copy Value (Attempt Type Conversion)". Please see the screenshots below for details.
