Hagoromo's Sharingan seems more like an unusual exception, since Hagoromo has a higher concentration of Otustsuki blood than any other human (except his brother Hamura, of course), as Hagoromo is 50% Otsutsuki, meaning that Hagoromo has special properties, powers, and rules for his dojutsu that other humans didn't have. The fact that he is the only person to naturally possess the Rinnegan, Sharingan, and the Byakugan in itself shows that he is a special case.
Considering that Hagoromo had the Sharingan, Rinnegan, and the Byakugan (and none of Hagoromo's other children did), it seems that the dojutsu rules are a bit different for a half-alien as opposed to his descendants who were 75% human and their alien blood only continued to become increasingly diluted. Eventually Hagoromo's descendants lost both of the Otsutsuki dojutsu, since Indra's descendants possessed neither Byakugan nor Rinnegan, but rather only the human mutations of the Sharingan and the Mangekyou remained.
Hagoromo appeared to be the first – and only – person capable of awakening the Rinnegan naturally, without injecting cells from another person into himself. The way that Hagoromo awakened his Rinnegan appears to be triggered similarly to how the Mangekyou Sharingan awakens, since Hagoromo's dojutsu responded to his feelings of grief over fatally injuring his brother, which is close to the requirements of grief and guilt that it would take to awaken the Mangekyou Sharingan, since it appears that Hamura would have died had not Hagoromo healed him first.
Although Hagoromo's dojutsu might operate a bit differently than the Uchiha, what is the same is that both his Sharingan and Rinnegan were awakened by negative emotions, namely grief and guilt, the same emotions that trigger the Sharingan. For those who believe in the so called "Curse of Hatred" of the Uchiha's Sharingan being awakened by negative emotions, that biological rule actually started long before Indra was even born, starting with Hagoromo awakening his dojutsu from negative emotions.
That is to say, either Hagoromo was under the so-called "Curse of Hatred" too (or he started it) or that it's simply a biological rule that the Sharingan awakens in reaction to feelings of grief and sadness. I guess it depends on whether you prefer the "spiritual laws rule the universe" theory of how the Sharingan awakens or prefer to see the Sharingan as a genetic trait whose awakening is rooted in biology (more or less, I mean the Sharingan is part of a magic system after all, so the "science" of it can only go so far), though I'll get more into that later.
Fans tend to assume that Hagoromo automatically awakened the Mangekyou Sharingan when he jumped ahead to the Rinnegan stage, gaining both Mangekyou and Rinnegan at the same time. I don't know that I would assume that, since Hagoromo's dojutsu operate a bit differently, and the Mangekyou might have been a mutation that started with Indra rather than Hagoromo. Which means there is a possibility that Hagoromo never had the Mangekyou, especially since we never see him use any of the Sharingan's abilities.
