The two dewberry trees you found are actually the same tree, the difference being whether it is currently day or night in Tibia. Daytime is 15 minutes past the hour until 45 after, while nighttime is 45 past the hour until 15 after. The Dewberry (green) appears during the day, while Bluish Dewberry Palm (blue) only appears at night.
https://tibia.fandom.com/wiki/Dewberry_Palm
https://tibia.fandom.com/wiki/Bluish_Dewberry_Palm
Harvesting fruit appears to be guaranteed, but only for certain trees. I could not find a consistent rule for what trees work and which ones don't outside of being able to harvest them in the moment. That said, Dewberry Palm (green) will only give dewberries, and Bluish Dewberry Palm will only give moon-touched dewberries when used.
I went and tested the cooldown for the fruits, and noticed that trees that could be gathered from had an effective cooldown of once per day/night cycle. The cooldowns are independent of both forms (Bluish and normal), so you can gather one of each fruit from the same tree almost immediately if both are off cooldown, can be gathered from (a daytime tree may be gatherable, but its nighttime counterpart may not), and timed at the change of day.