Can I demand to remove with any database (eg. web directory or domains directory) placed there data about my own website (eg. blog), by legal means and on what legal basis?
As noted by NIV, your state name (or the name of your country) and more details about the databases are needed to help you determine if what you wish to do is doable.
Is it for privacy reasons, or for other reasons, that you wish to have data about your blog/website removed from databases?
Here is a link to (older) information on Data Protection, from Privacy International. The information in the link refers to Obama and, of course with Trump, privacy rights have already changed in some notable (and somewhat disturbing) ways.
https://www.privacyinternational.org/node/44