OBJECT_ID of object in another database - how to find database ID or name/fully qualified object name?

You should be able to do this:

SELECT
   name
FROM
    AnotherDB.sys.objects   --changes context
WHERE
    object_id = OBJECT_ID('AnotherDB.ASchema.ATable')

This is what you effectively do with OBJECT_ID('AnotherDB.ASchema.ATable')

This means that you could rely on dbname.sys.objects and avoid confusion with metadata functions.

Note: the new Catalog views are designed to be used and not change from version to version, as per the link. In the old days, it was consider bad practice to use system tables but the stigma still remains. So, you can safely rely on sys.objects rather that the metadata functions.

Do I understand it correctly that you want the db id of AnotherDB?

SELECT *
FROM    master..sysdatabases
WHERE   name = 'AnotherDB'

Otherwise, you can USE other db's in dynamic SQL if it helps:

DECLARE @SQL    NVARCHAR(MAX)
,   @objId  INT

SET @SQL = N'
    USE AnotherDB

    SELECT  @id = OBJECT_ID(''customer'')
'

EXEC SP_EXECUTESQL @SQL
    ,   N'@id INT OUTPUT'
    ,   @id = @objId OUTPUT

SELECT  @objId

OR Execute SP's in other dbs with:

EXEC AnotherDB.dbo.ProcedureName 
      @paramX = ...
,     @paramY = ...

Take a look at the PARSENAME function in TSQL - will allow you to pull out any of the 4-part portions of a fully (or non-fully) qualified name. For the database in your example:

select parsename('AnotherDB.ASchema.ATable',3)

returns:

AnotherDB

select parsename('AnotherDB.ASchema.ATable',2)

returns:

ASchema

If non-fully qualified, you'll get null results if you ask for the portion of a name that isn't included in the string:

select parsename('ASchema.ATable',3)

returns:

NULL