Yes and no, depending on what kind of stylus you mean.
All capacitive touch-screens (the usual kind on phones) will "support" a stylus that emulates the capacitance of a human finger, with no additional components needed. It's not very precise, however – although better than a finger, but I'm not sure if it would be suitable for drawing.
What the video talks about is a different kind of stylus – one that does require a special screen with a digitizer that's just for the stylus (not necessarily a touch screen, although it can be both). For example, many Wacom drawing tablets are not touch-sensitive at all (for that matter, many of them are not even displays) – though, their page says that their newer devices use a kind of capacitive sensing that works with both pens and fingers, but apparently still more advanced than that of regular touchscreens.