The way the DMCA works ISPs or any other service hosting third-party content must comply with
every takedown notice immediately, regardless of it's legitimacy, if they want to continue to enjoy the protections of the DMCA. (meaning, unless they want to accept liability for the copyright of every piece of content on their servers). So yes, it absolutely does work that way: you can send a takedown notice to anybody for any reason and they must comply with it.
The logic is that DMCA notices are sent under penalty of perjury, and people aren't going to risk perjuring themselves just to improve their SEO or take down a blog post they don't like. Unfortunately, the courts seem willing to accept excuses like "the automated process i use to send out DMCA takedown notices got a little overzealous" as legitimate, so DMCA takedowns are now pretty much just a free-for-all for anybody who wants anything taken down.