JIRA itself stores only minimal user data (username, name, email, preferences). Since JIRA 3.7, you are able to store data in user 'properties'. You can store each customer detail as a separate user property, or create a wiki page for the customer and link to that instead. You could do this in Confluence by adding a Customer space and creating a page for every customer with their details. Then in JIRA, add a user property containing the link to that customer's page.
An alternative is to store user data in an LDAP server such as Active Directory or OpenLDAP. You can then authenticate users in JIRA against their LDAP password (see Configuring LDAP) and link to their full LDAP profile if available online.
There is also an open feature request for improved user properties at JRA-6354. You may wish to sign up for a user account and vote or comment to help influence our product roadmap.