As I was looking around how to fix the problem for quiet a while and did not find my solution anywhere:

There is also the option to set "type": "module" for a background script as described here: https://developer.mozilla.org/en-US/docs/Mozilla/Add-ons/WebExtensions/manifest.json/background

so in manifest.json it would be as follows:

"background": {
        "scripts": ["background.js"],
        "type": "module"
    }