Ok I managed to solve the problem :OO Leaving the answer to someone that may have same problem in the future :D
In my manifest file I had to change background.service_worker to background.scripts.
Apparently service_worker runs on the separate thread and has limited browser context hence no access to the cookies
If you use background.scripts instead it will make extension generate persistent background page that has a possibility to pickup the cookies :D Everything works out of the box now :D
Topic:
Safari & Web
SubTopic:
General
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