From my understanding, the God Tree produces chakra fruit once every thousand years by absorbing the natural chakra in the earth. After arriving on Earth, Kaguya bit into the chakra fruit and gained immeasurable power, using it to liberate all war and conflict.
So if the fruit was produced by the God Tree, does this mean that Kaguya ate a fruit of natural chakra?
Every individual on Earth has chakra. That chakra initially originated from the Shinju (a.k.a Ten-Tails). However, Natural chakra (or Senjutsu chakra) is the ever-present unlimited energy on the planet. Its usage depends on the shinobi's ability to draw and maintain it during battle.
Kaguya's chakra is not the same as natural chakra but it is the same chakra that is present in all the shinobi or rather, the original form of chakra present in all the shinobi's.
Hagoromo's chakra is a combination of the Ten-Tails as well as Senjutsu chakra. Which was why, when he shared it to Naruto and Sasuke, they were able to inflict damage to Madara, who was the host of Ten-Tails at the moment.
