psql: FATAL: role "postgres" does not exist

NOTE: If you installed postgres using homebrew, see the comments from @user3402754 and @originalhat below.

Note that the error message does NOT talk about a missing database, it talks about a missing role. Later in the login process it might also stumble over the missing database.

But the first step is to check the missing role: What is the output within psql of the command \du ? On my Ubuntu system the relevant line looks like this:

                              List of roles
 Role name |            Attributes             | Member of 
-----------+-----------------------------------+-----------
 postgres  | Superuser, Create role, Create DB | {}        

If there is not at least one role with superuser, then you have a problem :-)

If there is one, you can use that to login. And looking at the output of your \l command: The permissions for user on the template0 and template1 databases are the same as on my Ubuntu system for the superuser postgres. So I think your setup simple uses user as the superuser. So you could try this command to login:

sudo -u user psql user

If user is really the DB superuser you can create another DB superuser and a private, empty database for him:

CREATE USER postgres SUPERUSER;
CREATE DATABASE postgres WITH OWNER postgres;

But since your postgres.app setup does not seem to do this, you also should not. Simple adapt the tutorial.