From a hydration standpoint, does drinking non-caffeinated tea, such as chamomile, count towards a persons daily recommended intake of water?

Yes: Decaffeinated beverages count. The reason that caffeinated beverages don't is that caffeine is a diuretic. Thus, your urinary losses would counter any fluid intake you gained by drinking the beverage. Hope this helps!

Likely: If it is decaffeinated and has no other tea leaves in it with substances that act as diuretics. If you are concerned have another glass of water.