Postgresql – How to check connected user on psql

Yes, SERIAL is the equivalent function.

CREATE TABLE foo (
    id SERIAL,
    bar varchar
);

INSERT INTO foo (bar) VALUES ('blah');
INSERT INTO foo (bar) VALUES ('blah');

SELECT * FROM foo;

+----------+
| 1 | blah |
+----------+
| 2 | blah |
+----------+

SERIAL is just a create table time macro around sequences. You can not alter SERIAL onto an existing column.

PostgreSQL since version 9.5 has UPSERT syntax, with ON CONFLICT clause. with the following syntax (similar to MySQL)

INSERT INTO the_table (id, column_1, column_2) 
VALUES (1, 'A', 'X'), (2, 'B', 'Y'), (3, 'C', 'Z')
ON CONFLICT (id) DO UPDATE 
  SET column_1 = excluded.column_1, 
      column_2 = excluded.column_2;

Searching postgresql's email group archives for "upsert" leads to finding an example of doing what you possibly want to do, in the manual:

Example 38-2. Exceptions with UPDATE/INSERT

This example uses exception handling to perform either UPDATE or INSERT, as appropriate:

CREATE TABLE db (a INT PRIMARY KEY, b TEXT);

CREATE FUNCTION merge_db(key INT, data TEXT) RETURNS VOID AS
$$
BEGIN
    LOOP
        -- first try to update the key
        -- note that "a" must be unique
        UPDATE db SET b = data WHERE a = key;
        IF found THEN
            RETURN;
        END IF;
        -- not there, so try to insert the key
        -- if someone else inserts the same key concurrently,
        -- we could get a unique-key failure
        BEGIN
            INSERT INTO db(a,b) VALUES (key, data);
            RETURN;
        EXCEPTION WHEN unique_violation THEN
            -- do nothing, and loop to try the UPDATE again
        END;
    END LOOP;
END;
$$
LANGUAGE plpgsql;

SELECT merge_db(1, 'david');
SELECT merge_db(1, 'dennis');

There's possibly an example of how to do this in bulk, using CTEs in 9.1 and above, in the hackers mailing list:

WITH foos AS (SELECT (UNNEST(%foo[])).*)
updated as (UPDATE foo SET foo.a = foos.a ... RETURNING foo.id)
INSERT INTO foo SELECT foos.* FROM foos LEFT JOIN updated USING(id)
WHERE updated.id IS NULL;

See a_horse_with_no_name's answer for a clearer example.