Website blocking is fully active in Safari via the FocusDragon extension. Other browsers (Chrome, Firefox, Brave, Arc, Tor, etc.) get force-quit while a block is active — that's intentional. Allowing them would let you sneak past the Safari extension.
Support for Chromium-based browsers is on the roadmap. In the meantime, if you rely on Chrome for work, add it as an exception, but know that defeats the purpose of a distraction block.