Unbranded builds are available from the continuous integration builds on treeherder.mozilla.org.
What is different compared to Nightly and Developer editions (which allow unsigned addons too)?
It is stable or release version that allows unsigned addons, unlike those two unstable/experimental ones.
Same version is released whenever stable Firefox is released.
Some modified addons can't be installed although I made necessary change in about:config?
You have to add id manually in manifest.json in corresponding (same level) heading (parentheses).
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How do the unbranded versions of Firefox work?
They work just like Firefox, with two differences: they have a setting to disable mandatory signature checks, and they don't have the Firefox name and logo (instead using a generic name and logo). These builds are available in the en-US locale only.
The Nightly and Developer Edition versions of Firefox have a preference to disable signature enforcement. There are also be special unbranded versions of Release and Beta that have this preference, so that add-on developers can work on their add-ons without having to sign every build. To disable signature checks, you will need to set the xpinstall.signatures.required preference to "false".
type about:config into the URL bar in Firefox
in the Search box type xpinstall.signatures.required
double-click the preference, or right-click and selected "Toggle", to set it to false.