The DMCA is the US's implementation of the 1996 treaties of the World Intellectual Property Organization (the bill's formal name in the House was actually "WIPO Copyright and Performances and Phonograms Treaties Implementation Act of 1998"). Most countries you'd want to run a startup in likely have similar laws -- 187 are members of WIPO. The worst thing a falsely filed notice can do is take down a webpage for under two weeks, until a counter-notice goes in effect, while opening the false filer up to criminal liability. The law includes penalties for such bad faith filings.
Really? I'm not aware of other countries that have a "send a letter and content must be removed and you can't be sued". What's (say) the UK's equivalant of the DMCA?
To maintain the benefits of limited liability for their users' actions, a service provider must "act expeditiously to remove or to disable access to the information concerned" upon "obtaining actual knowledge or awareness of illegal activities". The easiest way to make a service provider aware of illegal activities is to notify them with evidence, like the identity of the copyright holder, identification of the original works, and attestation that the copyright holder did not authorize the copies, which are the exact content of a DMCA notice.
So, send a letter (provide awareness of illegal activity), remove the content, and the host can't be sued. That goes for all EU member states; the deadline to implement this directive into local law was back in 2002. Five non-EU-member states implemented it as well. That's in addition to the nearly 100 signatories to the WIPO treaty.
It's true that the format and timeframes of the DMCA system are more formal than the EU directive's requirements, but no matter what part of the world you look in, almost every developed country has formalized such a system in order to limit liability of content hosts for their users' action so long as they take down material under certain conditions. Some of the EU countries did formalize notice-and-takedown procedures of their own even though it wasn't required.