I have installed Ubuntu 16.04 LTS in my new Laptop (Dell Vostro 3568) about a week ago. My right-click is working same as the left click.
I am attaching a screenshot of the settings. I tried various options, but no result.

I too faced the same problem initially. After trying out a number of options, I realized that the left and the right buttons serve the same purpose. Maybe, it is a manufacturing defect. To get along, you can fix the problem by setting right as the primary button. This way you can use the touchpad for selecting objects and any of the two buttons for getting the pop-menu for these objects.