How to execute single sql query in multiple databases

If I guess you have all databases are in same structure and every time you run script to update something, the script basically same and you just run that same script one by one for each customer.

If the above case is true, you can use CURSOR to produce a Loop between your all databases and Execute necessary script to serve your purpose.

Note: This is not the solution, Just Idea.

--The first step will be creating a Table variable 
--where you will INSERT all your database names 
--for a further loop as below- 

DECLARE @DbName VARCHAR(200)
DECLARE @DatabaseList TABLE (DbName VARCHAR(200))

INSERT INTO @DatabaseList (DbName) VALUES('db_name_1')
INSERT INTO @DatabaseList (DbName) VALUES('db_name_2')
--.......................
INSERT INTO @DatabaseList (DbName) VALUES('db_name_50')

--Now you can use CURSOR to generate the loop 
--and execute your required script as shown below

DECLARE db_cursor CURSOR FOR 
SELECT DbName FROM @DatabaseList

OPEN db_cursor  
FETCH NEXT FROM db_cursor INTO @DbName  

WHILE @@FETCH_STATUS = 0  
BEGIN  

    --HERE You need to write your script That you
    --Execute for all your database. I have added
    --a sample script where I guess you updating 
    --certain tables in your all database WHERE ID = 1

    -- You can see the Database Name inserted in the 
    -- Script Dynamically from the Loop

    EXEC ('UPDATE '+@DbName+'.dbo.<Your_table_Name_Here> 
           WHERE ID=1')

    --END OF Dynamic Part

    FETCH NEXT FROM db_cursor INTO @DbName 
END 

CLOSE db_cursor  
DEALLOCATE db_cursor