Hagoromo's Rinnegan doesn't work the same as it does for the Uchiha clan, as his dojutsu was awakened by a combination of grief and guilt caused by a traumatized emotional response when he was forced to stab his twin brother in the heart in order to defend himself. Thus Hagoromo's Rinnegan was essentially triggered in the same way that a Sharingan is awakened, and arguably triggered by an emotional response more equivalent to that which awakens the Mangekyou Sharingan. So in Hagoromo's case (that is, the case of an Otsusuki/human hybrid), the conditions for awakening his Rinnegan was either (1) The same requirements as for a Mangekyou Sharingan but with a different outcome (Rinnegan instead of a Mangekyou) or (2) the Rinnegan simply requires intense negative emotion, same as with the Uchiha clan.
Oops! This image does not follow our content guidelines. To continue publishing, please remove it or upload a different image.
After the Sage of Six Paths, only Indra's reincarnations can awaken the Rinnegan. Which means that at any given time, only one person on earth has the latent potential to awaken Rinnegan.
For Indra's reincarnation to awaken Rinnegan, the criteria is this: Indra's reincarnation must have Asura chakra to awaken the Rinnegan. That's pretty much it. Indra and Asura chakra must be combined together for a Rinnegan awakening to be possible.
Interestingly, that makes Rinnegan the only version of Sharingan that doesn't respond to emotional pain or distress, but to more technical factors. Instead, one must inject Asura chakra into an Indra reincarnation somehow; it's not about emotions since that's not what the Rinnegan's awakening is linked to.
To make it even more confusing, Hagoromo's Rinnegan was linked to awakening in response to his negative emotions, yet Indra's reincarnations aren't. But as I've said before, Hagoromo is the exception to the rule due to being half-Otsutsuki and he abides by different rules than the rest of the Uchiha clan, who have a more diluted percentage of Otsutsuki blood running through their veins.
Madara is the first known user to awaken Rinnegan – not that that necessarily means much, since Madara and Sasuke are the only known reincarnations of Indra despite there likely being several hundreds of them across history. Madara may have been the first person since Hagoromo to awaken the Rinnegan, but that is unclear due to how little is known of history, so we don't really know (Sasuke hadn't even heard of Madara before despite Madara being a prominent and influential political and historical figure who was probably Sasuke's direct ancestor, not to mention being a co-founder of the village where Sasuke lived, so that in itself should tell you that the shinobi in Naruto are terrible historians and quickly forget history and apparently aren't interested in passing it down). Maybe a lot of Indra's reincarnations awakened the Rinnegan, maybe a few did, and maybe none of them did until Madara and Sasuke.
Madara awakened his Rinnegan by grafting Hashirama's cells into himself. Interestingly, the Rinnegan's awakening was not instantaneous for Madara, but rather something that was delayed and took a while to finally awaken.
Oops! This image does not follow our content guidelines. To continue publishing, please remove it or upload a different image.
Similarly, Sasuke awakened his Rinnegan when he received Hashirama's cells - and with it Asura's chakra – from Kabuto when Kabuto healed Sasuke during the war.
Oops! This image does not follow our content guidelines. To continue publishing, please remove it or upload a different image.
The precise details of Sasuke's awakening are a bit hazy, considering that it's unclear if Hagoromo had any involvement in Sasuke's Rinnegan awakening, since he granted Naruto and Sasuke both half of his power, though I think that power was more the ability to seal Kaguya away. So it's unclear if Hagoromo's power helped Sasuke's Rinnegan to awaken in any way, like perhaps speeding up the awakening of Sasuke's Rinnegan so it didn't take as long to awaken as Madara's did. Or perhaps the reason Sasuke's Rinnegan awakening was so much faster is because it was only awakened in one eye rather than both.
Oops! This image does not follow our content guidelines. To continue publishing, please remove it or upload a different image.
In any case, both Madara and Sasuke awakened their Rinnegan by receiving Asura's chakra and that responded to their Indra chakra. Those two halves became a whole that allowed them to awaken the Rinnegan like Hagoromo had. This can't happen in reverse with an Uzumaki, Senju, or any of Asura's descendants awakening the Rinnegan by transplanting Indra's chakra into themselves because Asura's descendants don't have the latent potential for a dojutsu; that's something exclusive to Indra's descendants.
Sasuke also cannot deactivate his Rinnegan for some reason, which is why it is left permanently on. He also seems unable to switch his Rinnegan eye to the regular Sharingan eye or Mangekyou pattern, so for some unknown reason, Sasuke has an unusual lack of control over his Rinnegan and isn't able to deactivate it. It is also unclear why Sasuke's Rinnegan has 6 tomoe, as compared to his predecessor Madara, whose Rinnegan had no tomoe.
I've often seen fans mock Sasuke for not resorting to the easy solution of using one of Pain's Six Paths abilities to resolve problems in Boruto, as fans often point out a specific ability that Sasuke's Rinnegan could easily replicate and complain that he didn't imitate Pain's finesse. While I agree that the writing is appallingly lackluster and disappointing when it comes to Sasuke's usage of his Rinnegan powers in Boruto (since he rarely ever uses it, and he doesn't use it well in the few instances when he does use it), I must somewhat dissent with such critiques, since that makes the unproven assumption that Sasuke's Rinnegan is capable of doing everything that Pain's Rinnegan did, which we don't know for a fact. The two Rinnegan might very well have entirely different abilities.
After all, if one advanced phase of the Sharingan, the Mangekyou Sharingan, grants users each abilities completely unique to them, it seems reasonable to assume that an even more advanced evolution of the Mangekyou Sharingan, the Rinnegan, might have unique abilities per user rather than having a standardized set of abilities for all Rinnegan. At least, from what I've observed, the Rinnegan of the Otsutsuki clan members seem to do different things and each Otsutsuki member seems to have slightly different abilities unique to their specific Rinnegan.
Or in simple terms, just like each Mangekyou Sharingan has a unique ability exclusive to that set of eyes, each Rinnegan might have a unique ability, even if the Rinnegan patterns aren't quite as diverse in design as the Mangekyou Sharingan (though they do differ in colors and number of tomoe and placement on the body).
From what I've garnered, the Rinnegan seem to possess different abilities from user to user, even if there are some shared abilities, like dimension traveling. We don't even know for sure if Madara's Rinnegan had the capability to travel through dimensions like Sauke's can, nor do we know if Pain was even capable of using Amenotijikara like Sasuke's Rinnegan.
Some of the Rinnegan's abilities are obviously similar, like Sasuke can use Planetary Devastation or Pain's Almighty Push, but it's unclear how far the similarities go as well as how far the differences go.
As for the number of Rinnegan's abilities, the Rinnegan comes with an insane amount of abilities and so many different powers that I can't even keep track of all of them, so I'm not even going to try to list them all (I'm sure Wikipedia can help with that, for readers who are interested).
