The Rinnegan is consistent of chakra, stated by Obito, and to awaken the Rinnegan, one must have obtained Hagaromo's chakra. They both have Hagaromo's chakra, they are just exhausted of the majority of it.
Hagoromo said it takes Indra and Ashura chakra to obtain his chakra which awakens the Rinnegan. Hagoromo gave Sasuke his Yin chakra which awakened Sasuke's Rinnegan but it doesn't mean he have the Sage's chakra as his own chakra since he never awakened that chakra as Madara did, the Sage chakra Sasuke and Naruto had got used up to seal Kaguya. That's why Sasuke told Naruto to use the hermit power and that he'll use the Sage's power as well which allowed him to use black Chidori.
The Sage chakra was separate from their own chakra, Sasuke and Naruto were both low on chakra so if they had the Sage chakra then they would have used it and neither one can make more of that chakra unless they take the other's cells to have both Indra and Ashura chakra. The Rinnegan have Sage power which is hard to control, the eye have sage chakra but you don't get chakra from the eye but put chakra into it to use it but it wasn't Obito's, Obito did not have have Ashura and Indra's chakra nor was it his eye.