A PC, notebook, tablet, phone, tv, glass running on Linux is still Linux. The user interface maybe different, but, the core OS is still the same, just like windows and Xbox. And that still consider the same family because the programming model is similar for hardware and software vendors to support it.
The vision is unified OS across hardware, but, not across devices and not about the experiences. Linux is getting more popular because that's what they are doing. You see tons of hardware vendor supporting Linux for this reason.
It certainly is important to maintain identity like WinPhone instead of WinMo. But you really don't have to transfer the same identity to a new device. You want the same experience on the same type of devices, but not trying to use a mouse in an automotive computer.