Postgresql ERROR: duplicate key value violates unique constraint when trigger insert
You're calling this

INSERT INTO organization VALUES ('Tivoli11','Tivoli organization',NULL,'F',NULL,NULL)


Which is triggering this.

INSERT INTO organization VALUES ('Tivoli1','Tivoli1 organization',NULL,'F',NULL,NULL)


Your primary key, organization_pkey, is on abbreviation. Here you have two Tivoli1 abbreviations. That triggers the exceptions which causes the trigger to fail, and the commit to abandon.

I have no idea what you want to do, but perhaps you need a consultant to audit your schema and workflow. It seems like you may be in over your head.

Re: "there is no Tivoli11"

Right,


You tried to insert it
That triggered a trigger trg_backup_row which you did not show in the question (please do)
trg_backup_row attempted another insert.
That attempted insert violated the unique constraint of the primary key.
That violation threw an exception.
That exception caused the transaction and the initial commit to rollback.


Transactions, even AFTER INSERT that throw exceptions cause the insert to rollback.