A Microsoft service that provides website owners with tools to monitor, analyze, and optimize their site's performance in Bing search results.
Hello Michał Milanowski,
Since the sign-in loop occurs in both Chrome and Edge, including private browsing, and you've already cleared browser data, further basic browser troubleshooting may not be productive.
Unfortunately, we cannot verify the underlying cause or restore access directly through the community forum. This appears to require investigation by the Bing Webmaster Tools support team rather than Microsoft account password troubleshooting. The official support channel is Bing Webmaster Support.