Hi @sql user , welcome to Microsoft Q&A forum.
DMS has the limitation to use primary keys on all tables to handle syncing to target database. However as a workaround we can temporarily set a primary key for the table for migration to continue. We can remove the primary key after data migration is complete. Same is mentioned in below article:
Known issues/migration limitations with online migrations from PostgreSQL to Azure DB for PostgreSQL
Other way is to use dump and restore method:
Migrate your PostgreSQL database by using dump and restore
Please let us know if this helps or else we can discuss further. If answer helps, you can mark it 'Accept Answer'.