Postgres inheritance and foreign keys or alternative

PostgreSQL inheritance doesn't quite work how you expect. Yes, you can see info from the child tables when querying the parent table, but this does not extend to foreign key relationships. The row "belongs" to the child, not the parent. The foreign key reference doesn't touch the child.

It's generally a bad idea to use inheritance in PostgreSQL except for specific cases like making a temporal system or enforcing naming conventions (like interfaces in OOP rather than state inheritance).

PostgreSQL inheritance can be very powerful, but it is generally overused in my opinion. There is already a solution (and cross-database compatible) that more closely follows the traditional relational model.

A better model that would do what you seem to want is the following:

CREATE TABLE "user" (
  user_id serial PRIMARY KEY, -- Or UUID or generated column in newer versions
  -- other fields that all "children" should share
);

CREATE TABLE user_child (
  user_id integer NOT NULL
    REFERENCES "user" (user_id) ON UPDATE CASCADE ON DELETE CASCADE,
  -- other fields specific to the child
);

CREATE TABLE homework (
  homework_id serial PRIMARY KEY,
  user_id integer NOT NULL
    REFERENCES "user" (user_id) ON UPDATE CASCADE ON DELETE RESTRICT,
  -- other fields specific to homework
);

The equivalent to your query for user_child is

SELECT u.user_id
FROM "user" AS u
  INNER JOIN user_child AS uc;

And to query user, both parent and child would still be

SELECT u.user_id
FROM "user" AS u;

Adding an inner join is pretty trivial and could be hidden behind a view. Now your foreign key reference to "user" will function correctly.