How to run the same query on all the databases on an instance?

Try this one -

    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

Output (for example, in AdventureWorks) -

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

How about something like this:

DECLARE c_db_names CURSOR FOR
SELECT name 
FROM sys.databases
WHERE name NOT IN('master', 'tempdb') --might need to exclude more dbs

OPEN c_db_names

FETCH c_db_names INTO @db_name

WHILE @@Fetch_Status = 0
BEGIN
  EXEC('
    INSERT INTO #report
    SELECT 
      ''' + @db_name + '''
      ,COUNT(*)
    FROM ' + @db_name + '..linkfile
  ')
  FETCH c_db_names INTO @db_name
END

CLOSE c_db_names
DEALLOCATE c_db_names

SELECT * FROM #report

Straight forward query

EXECUTE sp_MSForEachDB 
        'USE ?; SELECT DB_NAME()AS DBName, 
        COUNT(1)AS [Count] FROM CUSTOMERS'

This query will show you what you want to see, but will also throw errors for each DB without a table called "CUSTOMERS". You will need to work out a logic to handle that.

Raj