Oracle row level security setup?
You could use a row-level security policy with a statement_type limited to update (or more likely update and delete, and maybe insert too). See the DMBS_RLS.ADD_POLICY docs for details.

Dummy scenario: a list of tasks, only task owners can modify their task.

create table owners(owner_id int primary key
                  , owner_name varchar2(10));
create table tasks(task_id int primary key
                 , owner_id int
                 , description varchar2(20)
                 , completion number);

insert into owners(owner_id, owner_name) values (1, 'Mat');
insert into owners(owner_id, owner_name) values (2, 'Mark');
insert into tasks(task_id, owner_id, description, completion)
       values (100, 1, 'Task for Mat', 0);
insert into tasks(task_id, owner_id, description, completion)
       values (200, 2, 'Task for Mark', 0);
commit;


The policy function:

create or replace 
function tasks_update_policy(schema varchar2, tab varchar2)
return varchar2
is
  owner_id  number;
begin
  select owner_id into owner_id 
    from owners
    where lower(owner_name) = lower(sys_context('userenv','session_user'));
  return 'owner_id = ' || owner_id;
exception
  when no_data_found then
    return '1=2';  -- deny unregistered users
end;


Implementing the policy:

begin
  dbms_rls.add_policy(object_schema =&gt; 'MAT'
        , object_name =&gt; 'TASKS'
        , policy_name =&gt; 'Tasks_update_policy'
        , policy_function =&gt; 'tasks_update_policy'
        , statement_types =&gt; 'update,delete,insert'   -- policy restriction
        , update_check =&gt; true);
end;
/


When logged in as myself:

SQL&gt; select * from mat.tasks;

   TASK_ID   OWNER_ID DESCRIPTION      COMPLETION
---------- ---------- -------------------- ----------
       100      1 Task for Mat          0
       200      2 Task for Mark         0

SQL&gt; update mat.tasks set completion = 20 where task_id = 100 ;

1 row updated.

SQL&gt; update mat.tasks set completion = 20 where task_id = 200 ;

0 rows updated.


When connected as 'Mark':

SQL&gt; insert into mat.tasks values (101, 1, 'More work for Mat', 0);
insert into mat.tasks values (101, 1, 'More work for Mat', 0)
                *
ERROR at line 1:
ORA-28115: policy with check option violation