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¿Equivalente a la función R cbind() en Oracle?

En Oracle, debe definir explícitamente algún valor para obtener datos ordenados; sin eso, podría tener resultados diferentes cada vez que ejecuta una consulta.

Con tus datos, esto:

WITH table1 AS
         (SELECT 1 rn, '1'            col1 FROM DUAL UNION
          SELECT 2 rn, '2'            col1 FROM DUAL UNION
          SELECT 3 rn, 'NO_PATTERN'   col1 FROM DUAL UNION
          SELECT 4 rn, 'RANDOM_STUFF' col1 FROM DUAL),
     table2 AS
         (SELECT 1 rn, 'aaa'     col2 FROM DUAL UNION
          SELECT 2 rn, '4'       col2 FROM DUAL UNION
          SELECT 3 rn, 'qwewqeq' col2 FROM DUAL UNION
          SELECT 4 rn, 'UUUUUU'  col2 FROM DUAL)
SELECT col1, col2, t1.rn
from (select  row_number() over (order by rn) as rn, col1 from table1 ) t1
       inner join 
     (select row_number() over (order by rn) as rn, col2 from table2 ) t2
     on (t1.rn = t2.rn)

da:

COL1         COL2            RN
------------ ------- ----------
1            aaa              1
2            4                2
NO_PATTERN   qwewqeq          3
RANDOM_STUFF UUUUUU           4

Sin un pedido explícito, por ejemplo, este

WITH table1 AS
         (SELECT '1'            col1 FROM DUAL UNION
          SELECT '2'            col1 FROM DUAL UNION
          SELECT 'NO_PATTERN'   col1 FROM DUAL UNION
          SELECT 'RANDOM_STUFF' col1 FROM DUAL),
     table2 AS
         (SELECT 'aaa'     col2 FROM DUAL UNION
          SELECT '4'       col2 FROM DUAL UNION
          SELECT 'qwewqeq' col2 FROM DUAL UNION
          SELECT 'UUUUUU'  col2 FROM DUAL)
SELECT col1, col2, t1.rn
from (select  rownum as rn, col1 from table1 ) t1
       inner join 
     (select rownum as rn, col2 from table2 ) t2
     on (t1.rn = t2.rn)

da

COL1         COL2            RN
------------ ------- ----------
1            4                1
2            UUUUUU           2
NO_PATTERN   aaa              3
RANDOM_STUFF qwewqeq          4