Tabla MySQL Inner Join basada en el valor de la columna

Para tener las estadísticas de todas las tablas, puede usar una UNION, con 2 o más selecciones, una para cada tabla:

( SELECT s.*
       , table1.title AS name      --or whatever field you want to show
  FROM stats s
    JOIN $tableName1 table1
      ON s.id = table1.id
  WHERE tableName = '$tableName1'
)
UNION ALL
( SELECT s.*
       , table2.name AS name      --or whatever field you want to show
  FROM stats s
    JOIN $tableName2 table2
      ON s.id = table2.id
  WHERE tableName = '$tableName2'
)
UNION ALL
( SELECT s.*
       , table3.lastname AS name      --or whatever field you want to show
  FROM stats s
    JOIN $tableName3 table3
      ON s.id = table3.id
  WHERE tableName = '$tableName3'
)
;

Usando la idea de Winfred con LEFT JOIN s. Produce diferentes resultados, p. cada campo de las otras tablas se muestra en su propia columna (y se producen muchos valores NULL).

SELECT s.*
     , table1.title      --or whatever fields you want to show
     , table2.name
     , table3.lastname   --etc
FROM stats s
  LEFT JOIN $tableName1 table1
    ON s.id = table1.id
      AND s.tableName = '$tableName1'
  LEFT JOIN $tableName2 table2
    ON s.id = table2.id
      AND s.tableName = '$tableName2'
  LEFT JOIN $tableName3 table3
    ON s.id = table3.id
      AND s.tableName = '$tableName3'
--this is to ensure that omited tables statistics don't appear
WHERE s.tablename IN
   ( '$tableName1'
   , '$tableName2'
   , '$tableName3'
   )
;