What does it mean to wake up with an extremely dry throat on a regular basis?

Mouth breathing. Mouth breathing is the more likely cause of a dry throat. This could be related to nasal congestion or other issues. Sleep apnea also comes to mind. Using oxygen or a CPAP machine can cause a dry throat.
Dry throat on waking. You're probably mouth breathing while you sleep; so your throat dries out. If it were a salivary gland problem, the dryness would occur all the time. Try sleeping on your side. If you're tired even after a night's sleep, see your doc for possible sleep apnea work-up.