What’s the PostgreSQL datatype equivalent to MySQL AUTO INCREMENT?

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.