Sql-server – AD groups and ms SQL security: is this default behavior

If I am understanding your question correctly, then yes the user will be able to run both applications with database connection issues.

If User1 is part of Windows Groups Group1 and Group2, and Group1 is mapped as a SQL Server login, as well as Group2 is mapped to a different SQL Server login. And if Group1 is mapped to a database user in Database1 and Group2 is mapped as a database user in Database2, then User1 should have access to both Database1 and Database2. And if Group1 and Group2 are both mapped to TheOnlyDatabase, then the user will have the accumulations of both database users' permissions.

This will return only one of the database users (as expected):

select user_name() as database_user_name

This will return 1 for both queries if the users is a member of both Windows Groups:

select is_member('Domain\Group1') as member_of_group_1

select is_member('Domain\Group2') as member_of_group_2

Instead of relying on SSMS (its flakey sometimes - you might have to refresh, etc), always use T-SQL.

Below script will provide you comprehensive result of what permissions you have set (filter out as per your need)

/*
****I cant find the reference, but very descriptive script(I have it in my repository for years now - very useful for auditing).****

Security Audit Report
1) List all access provisioned to a sql user or windows user/group directly 
2) List all access provisioned to a sql user or windows user/group through a database or application role
3) List all access provisioned to the public role

Columns Returned:
UserName        : SQL or Windows/Active Directory user cccount.  This could also be an Active Directory group.
UserType        : Value will be either 'SQL User' or 'Windows User'.  This reflects the type of user defined for the 
                  SQL Server user account.
DatabaseUserName: Name of the associated user as defined in the database user account.  The database user may not be the
                  same as the server user.
Role            : The role name.  This will be null if the associated permissions to the object are defined at directly
                  on the user account, otherwise this will be the name of the role that the user is a member of.
PermissionType  : Type of permissions the user/role has on an object. Examples could include CONNECT, EXECUTE, SELECT
                  DELETE, INSERT, ALTER, CONTROL, TAKE OWNERSHIP, VIEW DEFINITION, etc.
                  This value may not be populated for all roles.  Some built in roles have implicit permission
                  definitions.
PermissionState : Reflects the state of the permission type, examples could include GRANT, DENY, etc.
                  This value may not be populated for all roles.  Some built in roles have implicit permission
                  definitions.
ObjectType      : Type of object the user/role is assigned permissions on.  Examples could include USER_TABLE, 
                  SQL_SCALAR_FUNCTION, SQL_INLINE_TABLE_VALUED_FUNCTION, SQL_STORED_PROCEDURE, VIEW, etc.   
                  This value may not be populated for all roles.  Some built in roles have implicit permission
                  definitions.          
ObjectName      : Name of the object that the user/role is assigned permissions on.  
                  This value may not be populated for all roles.  Some built in roles have implicit permission
                  definitions.
ColumnName      : Name of the column of the object that the user/role is assigned permissions on. This value
                  is only populated if the object is a table, view or a table value function.                 
*/

--List all access provisioned to a sql user or windows user/group directly 
SELECT  
    [UserName] = CASE princ.[type] 
                    WHEN 'S' THEN princ.[name]
                    WHEN 'U' THEN ulogin.[name] COLLATE Latin1_General_CI_AI
                 END,
    [UserType] = CASE princ.[type]
                    WHEN 'S' THEN 'SQL User'
                    WHEN 'U' THEN 'Windows User'
                 END,  
    [DatabaseUserName] = princ.[name],       
    [Role] = null,      
    [PermissionType] = perm.[permission_name],       
    [PermissionState] = perm.[state_desc],       
    [ObjectType] = obj.type_desc,--perm.[class_desc],       
    [ObjectName] = OBJECT_NAME(perm.major_id),
    [ColumnName] = col.[name]
FROM    
    --database user
    sys.database_principals princ  
LEFT JOIN
    --Login accounts
    sys.login_token ulogin on princ.[sid] = ulogin.[sid]
LEFT JOIN        
    --Permissions
    sys.database_permissions perm ON perm.[grantee_principal_id] = princ.[principal_id]
LEFT JOIN
    --Table columns
    sys.columns col ON col.[object_id] = perm.major_id 
                    AND col.[column_id] = perm.[minor_id]
LEFT JOIN
    sys.objects obj ON perm.[major_id] = obj.[object_id]
WHERE 
    princ.[type] in ('S','U')
UNION
--List all access provisioned to a sql user or windows user/group through a database or application role
SELECT  
    [UserName] = CASE memberprinc.[type] 
                    WHEN 'S' THEN memberprinc.[name]
                    WHEN 'U' THEN ulogin.[name] COLLATE Latin1_General_CI_AI
                 END,
    [UserType] = CASE memberprinc.[type]
                    WHEN 'S' THEN 'SQL User'
                    WHEN 'U' THEN 'Windows User'
                 END, 
    [DatabaseUserName] = memberprinc.[name],   
    [Role] = roleprinc.[name],      
    [PermissionType] = perm.[permission_name],       
    [PermissionState] = perm.[state_desc],       
    [ObjectType] = obj.type_desc,--perm.[class_desc],   
    [ObjectName] = OBJECT_NAME(perm.major_id),
    [ColumnName] = col.[name]
FROM    
    --Role/member associations
    sys.database_role_members members
JOIN
    --Roles
    sys.database_principals roleprinc ON roleprinc.[principal_id] = members.[role_principal_id]
JOIN
    --Role members (database users)
    sys.database_principals memberprinc ON memberprinc.[principal_id] = members.[member_principal_id]
LEFT JOIN
    --Login accounts
    sys.login_token ulogin on memberprinc.[sid] = ulogin.[sid]
LEFT JOIN        
    --Permissions
    sys.database_permissions perm ON perm.[grantee_principal_id] = roleprinc.[principal_id]
LEFT JOIN
    --Table columns
    sys.columns col on col.[object_id] = perm.major_id 
                    AND col.[column_id] = perm.[minor_id]
LEFT JOIN
    sys.objects obj ON perm.[major_id] = obj.[object_id]
UNION
--List all access provisioned to the public role, which everyone gets by default
SELECT  
    [UserName] = '{All Users}',
    [UserType] = '{All Users}', 
    [DatabaseUserName] = '{All Users}',       
    [Role] = roleprinc.[name],      
    [PermissionType] = perm.[permission_name],       
    [PermissionState] = perm.[state_desc],       
    [ObjectType] = obj.type_desc,--perm.[class_desc],  
    [ObjectName] = OBJECT_NAME(perm.major_id),
    [ColumnName] = col.[name]
FROM    
    --Roles
    sys.database_principals roleprinc
LEFT JOIN        
    --Role permissions
    sys.database_permissions perm ON perm.[grantee_principal_id] = roleprinc.[principal_id]
LEFT JOIN
    --Table columns
    sys.columns col on col.[object_id] = perm.major_id 
                    AND col.[column_id] = perm.[minor_id]                   
JOIN 
    --All objects   
    sys.objects obj ON obj.[object_id] = perm.[major_id]
WHERE
    --Only roles
    roleprinc.[type] = 'R' AND
    --Only public role
    roleprinc.[name] = 'public' AND
    --Only objects of ours, not the MS objects
    obj.is_ms_shipped = 0
ORDER BY
    princ.[Name],
    OBJECT_NAME(perm.major_id),
    col.[name],
    perm.[permission_name],
    perm.[state_desc],
    obj.type_desc--perm.[class_desc]