Insert data into multiple tables using one SQL query

The data is too dissimilar to cascade insertions, but if it was more regular (e.g. if your colours would be one for each setting rather than the three of them for the same setting, whilst there's one user setting for each setting) then it would make more sense. It can be done, though, like this (although it is quite silly):

CREATE temp TABLE SETTING (id int, description varchar(16), constrained varchar(5), data_type varchar(12), min_value varchar(6) NULL , max_value varchar(6) NULL);
CREATE TABLE ALLOWED_SETTING_VALUE (id int, setting_id int, item_value varchar(7), caption varchar(6));
CREATE TABLE USER_SETTING (id int, user_id int, setting_id int, allowed_setting_value_id varchar(6) NULL, unconstrained_value varchar(6) NULL);

-- the following is a single statement that inserts all 9 records
with
  allowed AS (
    INSERT INTO ALLOWED_SETTING_VALUE (id, setting_id, item_value, caption)
    VALUES (123, 10, '#0000FF', 'Blue'), (124, 10, '#FFFF00', 'Yellow'), (125, 10, '#FF00FF', 'Pink')
    RETURNING *),
  data(id, description, constrained, data_type, min_value, max_value, us_id, user_id, allowed_setting_value_id, unconstrained_value) as (
    VALUES
        (10, 'Favourite Colour', 'true', 'alphanumeric', NULL, NULL, 5678, 234, (select id::text from allowed where caption = 'Yellow'), NULL),
        (11, 'Item Max Limit', 'false', 'integer', '0', '9001', 7890, 234, NULL, '100'),
        (12, 'Item Min Limit', 'false', 'integer', '0', '9000', 8901, 234, NULL, '1')),
  settings as (
    INSERT INTO SETTING (id, description, constrained, data_type, min_value, max_value)
    SELECT id, description, constrained, data_type, min_value, max_value
    FROM data
    RETURNING *)
INSERT INTO USER_SETTING (id, user_id, setting_id, allowed_setting_value_id, unconstrained_value)
SELECT d.us_id, d.user_id, s.id, d.allowed_setting_value_id, d.unconstrained_value
FROM settings s
JOIN data d ON (d.id = s.id);

One way to have the whole thing running as if it was a single statement is enclosing it in a DO command:

DO $do$ BEGIN
  ... the whole script here ...
END; $do$;