search history via router?

No—but why not? Let's take a closer look.

This is an old URL:

 http://youtu .be/dQw4w9WgXcQ

Notice how it starts with http:// . This stands for HyperText Transfer Protocol —but that's not important right now; what matters is that a new URL looks like this:

 https://youtu .be/dQw4w9WgXcQ

As we can see, an s has been added. The s stands for Secure. So what makes this so much more secure?

Well, without the S, all intermediaries can see the full URL—the router, the ISP, etc.—but with the S, they can only see the basic domain you were on. The rest of the URL is then known only to that site.

Since all major search engines, including Google, Bing, DuckDuckGo, etc., use HTTPS, this means your router can only see that you were on Google (or Bing, or whatever), not what you searched for there. So, the only thing you can do is figure out how much you search per day and which search engine you use—but not what you search for.

Edit: The spaces in the URL are there to prevent Gutefrage from removing the links to embed automated previews. Please, Gutefrage, at least in code blocks, don't do that.