PostgreSQL UPDATE with conditional

A CASE expressions returns a single value it's not a replacement for an assignment.

Your current THEN expressions all return a boolean expression that compares the (current) value of the column country_code with the constant on the right hand side. You need to remove the comparison with the target column in your CASE expression:

UPDATE vaccine 
  SET country_code = CASE
                      WHEN vaccine_country='Germany' THEN 'DE'
                      WHEN vaccine_country='United States' THEN 'US'
                      WHEN vaccine_country='China' THEN 'CN'
                      WHEN vaccine_country='Turkey' THEN 'TR'
                      WHEN vaccine_country='Russia' THEN 'RU'
                    END

Or a bit shorter:

UPDATE vaccine 
  SET country_code = CASE vaccine_country
                      WHEN 'Germany' THEN 'DE'
                      WHEN 'United States' THEN 'US'
                      WHEN 'China' THEN 'CN'
                      WHEN 'Turkey' THEN 'TR'
                      WHEN 'Russia' THEN 'RU'
                    END

Another option is to provide the mapping with a VALUES clause instead of a CASE expression:

UPDATE vaccine 
   SET country_code = t.country_code
FROM ( 
  values 
     ('Germany', 'DE'), 
     ('United States' , 'US'),
     ('China' , 'CN')
     ('Turkey' , 'TR')
     ('Russia' , 'RU')
) AS t(country, country_code)
where vaccine.country = t.country;

Note that for rows where the vaccine_country does not contain any of the listed countries, the country_code will be set to NULL which might or might not be what you want.