Touchscreen controllers need to be specialized because the mobile phone always need to detect touch events at very high frequency to support sensitive touching experience. So it's a always on process. Making it a separate processor will help improve the performance of touch detection and also save more battery for the phone.
Personally, I think touchscreen is not always-on functionality. ASIC is used here for a special chip can save a lot of processing power (process the low resolution picture to get potential interesting points rather than using general CPU) and bandwidth (pass points' positions to general CPU rather than the raw input image).
An always-on functionality is something like "Google now" or "Hey Siri" (in new iPhone) that works even when phone is in your pocket. That means the device is always listening for the event and thus the functionality is always-on.
I agree with @Fantasy since most touchscreen phones need to be 'woken up' before they can recognize the touches which makes it not always on.