Table is specified twice, both as a target for 'UPDATE' and as a separate source for data in mysql

This is a typical MySQL thing and can usually be circumvented by selecting from the table derived, i.e. instead of

FROM manager AS m2

use

FROM (select * from manager) AS m2

The complete statement:

UPDATE manager
SET status = 'Y'
WHERE branch_id IN
(
  select branch_id
  FROM (select * from manager) AS m2
  WHERE (branch_id, year) IN
  (
    SELECT branch_id, year
    FROM branch_master
    WHERE type = 'finance'
  )
);

Try to use the EXISTS operator:

UPDATE manager as m1 
SET m1.status = 'Y'
WHERE EXISTS (SELECT 1
              FROM (SELECT m2.branch_id             
                    FROM branch_master AS bm
                    JOIN manager AS m2
                    WHERE bm.type = 'finance' AND
                        bm.branch_id = m2.branch_id AND
                        bm.year = m2.year) AS t
              WHERE t.branch_id = m1.branch_id); 

Note: The query uses an additional nesting level, as proposed by @Thorsten, as a means to circumvent the Table is specified twice error.

Demo here