What you're seeing is, in fact, the history from the web browser. Presumably Safari in your case. Finder itself doesn't have a 'History' menu option. And in your question you say "appears (when you open a web browser)" and that's the key bit of information.
The menu bar in OS X is shared among all applications. What is shown in the menu bar, that thin bar that runs along the top of your screen, changes depending on which application is in the foreground. The position of the menu bar never changes: it's always at the top of the screen. But the menu items in the menu bar change depending on what application is in the foreground.
For example, when Finder is in the foreground I see:
And when I switch to Safari I see:
Similarly with Chrome in the foreground I see:
So that 'History' item menu you're referring to is the browsing history for whatever browser you happen to be using. To clear the browsing history for...
From the menu bar select: History > Clear History...
Press Cmd-Y to open the History tab in Chrome and click the
Clear browsing data... button on that page.
Click the menu button and select History > Clear Recent History....