A common solution is based on this logic: replace the search string with an empty string and divide the difference between old and new length by the length of the search string
(CHAR_LENGTH(name) - CHAR_LENGTH(REPLACE(name, 'substring', '')))
/ CHAR_LENGTH('substring')
Hence:
UPDATE test."user"
SET result =
(CHAR_LENGTH(name) - CHAR_LENGTH(REPLACE(name, 'o', '')))
/ CHAR_LENGTH('o');