In a sense, yes there are two different clipboards (though X11 calls them "selections", and only one of the selections is called “the clipboard”).
In all likelihood, you are thinking of the Primary Selection (an understandable mistake considering that it functions much more like one on Linux distros than it does under Windows or Mac—though, for the record, the same keyboard shortcuts can typically be used to interact with the Primary Selection under Windows and Mac as they are under Linux).
The clipboard is typically controlled via the Ctrl+{c,v} key bindings for Copy and Paste (respectively). The Primary Selection, on the other hand, has several means of interaction. With the mouse, selecting anything automatically places it in the Primary Selection. Then, one can typically middle-click to Paste from it. One can also use the keyboard: {Ctrl,Shift}+Insert may be used for Copy and Paste (again, respectively) from/to the Primary Selection.
Oddly, it seems X's clipboard is not supported by every application. And, though it may seem unfortunate, there is no way to unify the Primary Selection and the Clipboard. However, there are two solutions for you.
Use the Primary Selection
It is almost universally available and (at least, it my humble opinion) is a fair bit more flexible.
Use an alternative clipboard manager
There are plenty of third-party clipboard managers out there which offer far more powerful functionality than the built-in.
Just for example, you could use xcmenu or xclip. There are, however, more alternatives than I can count.
A while after writing this answer, I stumbled upon a clipboard manager called clipit
. Among other features that can be quite handy, clipit
offers the ability to synchronize the Primary Selection and the X Clipboard. This is functionally equivalent to unifying the two selections and has been really nice to have; I imagine it is exactly what you are looking for.