I'm playing the Elder Scrolls Online right now, and I'm trying to understand some of the background of the Amulet of Kings. Specifically I'm confused about the relationship between dragonborn, the descendants of Alessia, and the Amulet of Kings. In the Trials of St. Alessia (http://www.uesp.net/wiki/Lore:Trials_of_St._Alessia) Akatosh says (emphasis mine):

But Tiber Septim wasn't a descendant of Alessia, right? So was either Akatosh or the author lying? Does that mean that the only requirement to wear the Amulet of Kings is to be dragonborn? Could Miraak, for example, have put on the Amulet of Kings because he was the first dragonborn, even though he was born an era before Alessia ever existed and made the pact with Akatosh?

In ESO none of the 3 faction leaders are either dragonborn or descendants of Alessia (as far as we know). So then the entire war for Cyrodil in ESO is completely misguided, because none of the contenders for the throne would be able to wear the Amulet of Kings and thereby protect Mundus from Oblivion? Even if Molag Bal is defeated and the planemeld is averted, there's nothing stopping any other daedric prince from walking right into Tamriel whenever they feel like it? Mehrunes Dagon could choose to start opening portals all over Tamriel tomorrow just like during the Oblivion crisis, if he wanted to?

Yes, the war for Cyrodiil is misguided in some sense, but the mechanics of the Amulet of Kings are rather contradictory and confusing. It's not a simple matter of being Alessia's heir; in fact, Skyrim added a book which addresses this directly:

The http://uesp.net/wiki/Lore:The_Book_of_the_Dragonborn. Even in universe, they don't truly get it. To further complicate the issue, the characters after the throne in ESO are actually aware of this. I don't know if you've played through the main quest yet - I don't own the game myself, but I've heard from others that there are unusual schemes and conspiracies relating to filling the Ruby Throne.

The factions also tend to have big reasons besides the protection of Mundus as a whole to take the reins of Cyrodiil. In fact you could argue that wider concerns are of less importance than seizing power, what with them being greedy rulers in a war-riven dark age.

The Dragonfires aren't the only thing that stops Oblivion from collapsing into Mundus. While this angle was understandably played up in TES IV, there are other things going on: Sotha Sil's pact with the Daedra (which forbids the summoning of a Daedric Prince), the metaphysical effects of the Towers, agreements and plots between the Princes themselves; in essence it's something of a manufactured plot point that Daedra invade in ESO and TES IV, but at the same time there are reasons which help to explain why it doesn't happen all the time. In ESO, it relates to the instability of the whole world; Tamriel is during the interregnum at a low point politically, spiritually, intellectually; all these things add up to being the right point for Molag Bal to make his appearance. Also, Mannimarco plays a big part.

That second possibility from the Book of the Dragonborn, seems to be refuted by what happened to Varen Aquilarius in ESO. He took the throne but still wasn't able to wear the Amulet (one would assume he tried to perform all the proper rituals) which led him to attempt Mannimarco's scheme to make him a dragonborn.

In response to your question about identifying Dragonborn, and the abilities of the Emperors and Empresses with that title: I think on the whole, yes. As strange as that may seem. But what the Dragonborn means to Nords (strongly associated with Thu'um prowess and dragon-like powers) is different to its relevance to Imperial culture (essentially, the divine right of rulers; the blessing is more important than what it actually "does"). I think it would also have been improper and dangerous for any Emperor to out as a storm-breathing draconic warlord of old, and anyway most of them would not have even known that's what the station of Dragonborn Emperor implied, much less how to use the power if needed.

I don't, however, think that there could be Dragonborn running around out there, because the Dragonborn means nothing without actions to identify her or him as such. If a Dragonborn arose at the same time as the throne was filled, that would very much look like Akatosh was challenging the existing rulers - if they were Septims or Remans - and removing his blessing. This is not the case with the Medes since they do not purport to be Dragonborn emperors at all.

But as for the last part: not all rulers of Cyrodiil are Dragonborn. It is not a strict requirement. In fact, the rituals were only properly codified in the days of the Reman emperors, although the human empire of the Cyrodiils had existed for more than a thousand years by that point. Notably the previous dynasty of rulers before ESO, the Akaviri potentate, did not wear the Amulet of Kings; they were not even "human" in a familiar sense. There is, however, a strong bias favouring the Dragonborn: if Varen had shown himself to be blessed by the Divines then he would have far more successfully promoted Cyrodiilic rule. This is part of the reason why Tiber Septim was so successful in his bid for the throne.

Further complicating this is that all the ancestors of the various Emperor lines have divine or supernatural origins and yet we have fairly ordinary people also taking the throne. Alessia was... Alessia, but she also married Morihaus, a half-bull son of Kyne and their son became Emperor after her. Reman I's father was Hrol who infamously "made love unto a hillock" who he thought was the spirit of Alessia
(maybe). Tiber Septum ascended eventually, but interestingly enough, none of his direct decedents hold the throne, instead it's the descendants of his brother who do. We also have Katariah, a full Dunmer, holding the throne for over 50 years as does her non-Septim related son.

So yeah, the dragonborn requirement for being emporer really doesn't make that much sense.

they were probably referring on the start of the dynasty not later on down the line as that's Dragonblood instead of Dragonborn.

I believe that there are two types of Dragonborns:

1. Those with the blessing of Akatosh who wear the Amulet of Kings, ie. Alessia and the Septim dynasty
2. The soul-eating, shouting ones such as Miraak, Reman, Tiber, and the LDB