インスタンス上のすべてのデータベースで同じクエリを実行するにはどうすればよいですか？

これを試してください-

    SET NOCOUNT ON;
    
    IF OBJECT_ID (N'tempdb.dbo.#temp') IS NOT NULL
       DROP TABLE #temp
    
    CREATE TABLE #temp
    (
          [COUNT] INT
        , DB VARCHAR(50)
    )
    
    DECLARE @TableName NVARCHAR(50) 
    SELECT @TableName = '[dbo].[CUSTOMERS]'
    
    DECLARE @SQL NVARCHAR(MAX)
    SELECT @SQL = STUFF((
        SELECT CHAR(13) + 'SELECT ' + QUOTENAME(name, '''') + ', COUNT(1) FROM ' + QUOTENAME(name) + '.' + QUOTENAME(@TableName)
        FROM sys.databases 
        WHERE OBJECT_ID(QUOTENAME(name) + '.' + QUOTENAME(@TableName)) IS NOT NULL
        FOR XML PATH(''), TYPE).value('text()[1]', 'NVARCHAR(MAX)'), 1, 1, '')
    
    INSERT INTO #temp (DB, [COUNT])              
    EXEC sys.sp_executesql @SQL
    
    SELECT * 
    FROM #temp t

出力（たとえば、AdventureWorks ）-

COUNT       DB
----------- --------------------------------------------------
19972       AdventureWorks2008R2
19975       AdventureWorks2012
19472       AdventureWorks2008R2_Live