Inserting date range arrays daterange[]
This would be legal syntax - using ISO format for dates (YYYY-MM-DD), which is always a good idea as it does not depend on the current setting of datestyle. Your example only works with datestyle = 'DMY'.
 '{&quot;[2018-03-19,2020-12-29)&quot;,&quot;[2018-06-21,2020-12-29)&quot;}'

You can always ask Postgres to show you:
test=# SELECT ARRAY ['[2018-03-19,2020-12-29)'::daterange ,'[2018-06-21,2020-12-29)'];
                         array                         
-------------------------------------------------------
 {&quot;[2018-03-19,2020-12-29)&quot;,&quot;[2018-06-21,2020-12-29)&quot;}
(1 row)

The error msg you report does not seem to be generated by Postgres.

argument formats can't be mixed


Are you, by any chance, using an outdated version of pgAdmin III? See:

pgAdmin argument formats can't be mixed