WHY YOU SHOULD CARE
Other search engines save your search history. Usually your searches are saved along with the date and time of the search, some information about your computer (e.g. your IP address, User agent and often a unique identifier stored in a browser cookie), and if you are logged in, your account information (e.g. name and email address).
With only the timestamp and computer information, your searches can often be traced directly to you. With the additional account information, they are associated directly with you.
Also, note that with this information your searches can be tied together. This means someone can see everything you've been searching, not just one isolated search. You can usually find out a lot about a person from their search history.
It's sort of creepy that people at search engines can see all this info about you, but that is not the main concern. The main concern is when they either a) release it to the public or b) give it to law enforcement.
Why would they release it to the public? AOL famously released supposedly anonymous search terms for research purposes, except they didn't do a good job of making them completely anonymous, and they were ultimately sued over it. In fact, almost every attempt to anonymize data has similarly been later found out to be much less anonymous than initially thought.
The other way to release it to the public is by accident. Search engines could lose data, or get hacked, or accidentally expose data due to security holes or incompetence, all of which has happened with personal information on the Internet.
Why would search engines give your search history to law enforcement? Simply because law enforcement asked for it, usually as part of a legal investigation. If you read privacy policies and terms of service carefully you will notice that they say they can give your information on court order.
This makes sense because they may be legally obligated to do so. However, search engines are not legally obligated to collect personal information in the first place. They do it on their own volition.
The bottom line is if search engines have your information, it could get out, even if they have the best intentions. And this information (your search history) can be pretty personal.
For these reasons, DuckDuckGo takes the approach to not collect any personal information. The decisions of whether and how to comply with law enforcement requests, whether and how to anonymize data, and how to best protect your information from hackers are out of our hands. Your search history is safe with us because it cannot be tied to you in any way.