More efficient SQL for different databases with same schema
There are various ways to write the query in shorter form. I'm not sure if they could improve performance (or make it worse!)

The best lines of attack if the query isn't efficient is to change it into UNION ALL (provided that the results are equivalent) and examine/optimize the parts separately, make sure indexes are used, etc.

One way to rewrite is to move the UNION or UNION ALL inside a derived table and then join:

SELECT a.col1, a.col2, a.col3, b.colX, b.colY, b.colZ
FROM  
    ( SELECT col1, col2, col3, DATE, someID
      FROM db_JAN.tab1    WHERE col1 = 'variable1' 
      UNION ALL
      SELECT col1, col2, col3, DATE, someID
      FROM db_FEB.tab1    WHERE col1 = 'variable1' 
      UNION ALL
      ---
    ) a
INNER JOIN db_OTHER.tabX b
ON (a.DATE = b.DATE AND a.someID = b.someID) ;