When using the designer for Mono for Android in Visual Studio,
the controls have a nice clean design but as soon as you run the app on a device (emulator or real)
the controls have a completely different style:

Is this because the designer uses a specific theme, if so which one,
or is there something else I have to do to get the same style on the device as in the designer?
You probably have the default theme set to Theme.Holo.Light in the designer. In the upper-right of the designer, you will see a button that allows you to select the theme. See the screenshot below.
If you need to change the theme to Holo light or just Holo, reference this post. You will need to make some themes resources and update your application manifest.