if the primary key exists, it should silently ignore the record if all the other columns match, but throw an error if they don't.

I'd think a unique index including any extra columns would make this version of your INSERT work:

drop   table if exists testtable;
create table           testtable (pk int primary key, foo text, bar te
+xt);
create unique index foobar_idx on testtable (pk, foo, bar);
INSERT INTO testtable (pk, foo, bar) VALUES (1, 'Hello', 'World') ON C
+ONFLICT (pk,foo,bar) DO NOTHING;
INSERT INTO testtable (pk, foo, bar) VALUES (1, 'Hello', 'World') ON C
+ONFLICT (pk,foo,bar) DO NOTHING;
INSERT INTO testtable (pk, foo, bar) VALUES (1, 'Hello', 'Quz'  ) ON C
+ONFLICT (pk,foo,bar) DO NOTHING;
[download]

which yields:

drop   table if exists testtable;
DROP TABLE
create table           testtable (pk int primary key, foo text, bar te
+xt);
CREATE TABLE
create unique index foobar_idx on testtable (pk, foo, bar);
CREATE INDEX
INSERT INTO testtable (pk, foo, bar) VALUES (1, 'Hello', 'World') ON C
+ONFLICT (pk,foo,bar) DO NOTHING;
INSERT 0 1
INSERT INTO testtable (pk, foo, bar) VALUES (1, 'Hello', 'World') ON C
+ONFLICT (pk,foo,bar) DO NOTHING;
INSERT 0 0
INSERT INTO testtable (pk, foo, bar) VALUES (1, 'Hello', 'Quz'  ) ON C
+ONFLICT (pk,foo,bar) DO NOTHING;
ERROR:  duplicate key value violates unique constraint "testtable_pkey
+"
DETAIL:  Key (pk)=(1) already exists.
[download]

Postgres 12.1 (but this works from 9.5.x onward)

Manual PostgreSQL 12.1: INSERT # SQL-ON-CONFLICT

update: The first INSERT should be the same, of course, fixed. Also, removed vestiges of the original bash-version.

update 2: Re 'MySQL vs. Postgres': I think open-sourced and completely-free PostgreSQL is a better/safer choice than proprietary Oracle MySQL. For one thing, over time Pg will get better -- even better than Oracle -- but how could Oracle ever allow cheapo MySQL to overtake its venerable and expensive Oracle database product?