This is definitely an issue with the SQL Server security, not Tableau. A few questions:
1) Can I assume you are using windows authentication in SQL Server?
2) Can you confirm that the user is added as database server user as well as a user within the specific database?
3) Is the user enabled?
4) Can the user connect via another method, such as directly connecting via SQL Server Management studio?
Following up on this one. Let us know how we can help further.