Your application is subject to DPI virtualization. Your app has not indicated to the system that it is aware of and supports high DPI display devices. The machine in question uses a font scaling of greater than 125%. That's the high DPI cut off point. Beyond that scaling, the system virtualizes the font scaling for non DPI aware apps. From the documentation:
Windows Vista introduced a feature called DPI virtualization, which provides a level of automatic scaling support to applications that are not DPI–aware. With this feature, Windows scales the size of the text and UI elements of applications that are not DPI-aware so that they are appropriately sized on high DPI settings without changes in the application. This prevents potential usability and readability issues that occur when applications render too small on high DPI screens.
In Windows Vista through Windows 8, this feature provides "virtualized" system metrics and UI elements to not DPI–aware applications, as if they are running at 96 DPI. The application then renders to a 96 DPI off-screen surface and DWM scales the resulting application window to match the DPI setting. For example, if the DPI display setting is 144, DWM scales the application's window by 150%, or 144/96.
A consequence of this is that the system fakes the screen dimensions. When your application is DPI virtualized the system reports the virtualized dimensions rather than the true dimensions.
Best practice is to declare your application to be high DPI aware, and scale the UI according to the user's font scaling preferences. Of course, this is going to be quite a change for you. Possibly not one that you wish to take one.
Another option is to ask the user to use a reduced font scaling. They probably won't be very keen on that either.
Yet another option would be to manifest your app to be high DPI aware and continue not to scale it. Then it would certainly run, but it would fail to respect the user's font scaling preferences. Again, I imagine that the user would be non-plussed.
If you don't currently manifest your app, then doing so now will result in it not being virtualized. And by that I mean app virtualization rather than DPI virtualization. Whilst you really should not still be running virtualized, you may run into trouble if you disable virtualization.
And Sertac suggests another option. Get the user to apply compatibility settings for your app to disable DPI virtualization. You could apply that to the app as it is today, without re-compiling and at least your user would be able to make progress.
Fundamentally, there is a problem with your user's machine setup. If it has 768 vertical pixels and around 140% scaling, that really is a 100% scaled equivalent of 550 pixels. That's not very many pixels. Your application is objecting because the screen is too small and, well, perhaps that really is the case.
This various answers here may be useful to you: How do I make my GUI behave well when Windows font scaling is greater than 100%