So I found out starting in Oblivion, that the past Emperors were Dragonborn and only allowed to wear the Amulet of Kings without it slipping off. In order for them to be Dragonborn, they must light the Dragonfires. But in Skyrim, it was mentioned that you need kill dragons to absorb their souls in order to become Dragonborn. And as the Dragonborn you can shout. But I was unsure about if the Emperors were able to do that, because it was never mentioned if the past Emperors could shout. Were they able to do things like the Dragonborn in Skyrim, or is this just a coincidence?
All past Emperors were considered Dragonborn, but we can only speculate on what they could actually do. That said, you did not need to be Dragonborn to shout, and being Dragonborn did not necessarily mean you could absorb dragon souls. Despite being considered "the first Dragonborn", we have word-of-god reference that St. Alessia could not absorb dragon souls.
Anybody can shout. Being a Dragonborn makes it considerably easier, but even then, you would still need to learn the language. Skyrim gives us more insight, throughout the main story;
The Greybeards can shout after dedicating their lives to learning to Thu'um, or speak in dragon. After 10 years of training with The Greybeards, Ulfric Stormcloak was able to use his shout to kill High King Torygg. We also know that these characters were not Dragonborn, as they still refer to the player character as the last Dragonborn.
Absorbing dragon souls
As IG_42 points out, during the reign of the Dragonborn Emperors, the dragons were extinct. The Emperors were simply never around to encounter a dragon soul. That said, we can gather further insight directly from one of the Elder Scrolls writers, suggesting that none of the Emperors could actually absorb dragon souls:
Merry Eyesore the Elk (AKA Michael Kirkbride : Word of God):
"Alessia didn't have the power to absorb dragon souls. Hers was a much more nuanced power: to dream of liberty and give it a name and on her deathbed make Covenant with the Aka-Tusk. Perhaps if you had read her histories of the Dragon War, this would be more clear."
Who could wear the Amulet of Kings?
"So I found out starting in Oblivion, that the past Emperors were Dragonborn and
only allowed to wear the Amulet of Kings without it slipping off."
Only direct descendants of St. Alessia could wear the Amulet of Kings.
It is alluded that there was a secondary requirement of being Dragonborn, but it is also accepted that descendants of St. Alessia would already be a Dragonborn. While we mostly only see an Emperor wearing the Amulet of Kings, in theory the lore would allow for any direct descendant to wear it.
"Akatosh, looking with pity upon the plight of men, drew precious blood from his own heart, and blessed St. Alessia with this blood of Dragons1, and made a Covenant that so long as Alessia's generations were true to the dragon blood, Akatosh would endeavor to seal tight the Gates of Oblivion
In token of this Covenant, Akatosh gave to Alessia and her descendants the Amulet of Kings and the Eternal Dragonfires of the Imperial City.
So long as the Empire shall maintain its worship of Akatosh and his kin, and so long as Alessia's heirs shall bear the Amulet of Kings, Akatosh and his divine kin maintain a strong barrier between Tamriel and Oblivion"
The Amulet of Kings - in-game lore book
In other words, you could wear the Amulet of Kings if you were simply a descendant of St. Alessia. The Emperor's son would be able to wear the Amulet of Kings without actually being the Emperor. In fact, there is further allusion that the Emperor must be a direct descendant because they can actually wear the Amulet of Kings, not the other way around. That is why it is so important, during the events of Oblivion, that Martin Septim inherits the role from his late father.
Why did the Emperors relight the Dragonfire?
In order for them to be Dragonborn, they must light the Dragonfires."
Lighting the Dragonfire were required to hold the barrier against the plane of Oblivion. They were not a requirement to become Dragonborn.
The Dragonfire acts as a ward to prevent invasion from the plane of Oblivion. The second the Emperor dies, the Dragonfire goes out. That is why the next Emperor must relight it - to reignite the barriers that prevent the Daedra of Oblivion from invading.
So long as you and your descendants shall wear the Amulet of Kings, then shall this dragonfire burn -- an eternal flame -- as a sign to all men and gods of our faithfulness. So long as the dragonfires shall burn ... I swear that my Heart's blood shall hold fast the Gates of Oblivion.
So long as the Blood of the Dragon runs strong in her rulers, the glory of the Empire shall extend in unbroken years. But should the dragonfires fail, and should no heir of our joined blood wear the Amulet of Kings, then shall the Empire descend into darkness, and the Demon Lords of Misrule shall govern the land.'
-- from the liturgy of the Re-Kindling of the Dragonfires
Trials of St. Alessia - in-game lore book
So you see, the Emperor did not relight the Dragonfire to become Dragonborn. They did so because they were under the understanding that if the Dragonfire was left unlit, the Daedra would invade, and their empire would be reduced to ruin.
How do you become a Dragonborn?
"But in Skyrim,
it was mentioned that you need kill dragons to absorb their souls in order to become Dragonborn."
Only a Dragonborn can absorb the soul of a dragon. It is not a pre-requisite for being Dragonborn, nor is it a natural capability of being Dragonborn.
In fact, being Dragonborn is mostly luck. The only requirements for being a Dragonborn is that the person is born with the blood and soul of a dragon, whilst otherwise retaining a mortal body.
Most of the recorded Dragonborn are the direct descendants of St. Alessia. As previously quoted, the 'inherited Dragon's blood' was a gift from Akatosh. We now know that St. Alessia could not actually absorb dragon souls. In fact, dialogue with The Greybeards leader Arngeir reveals that a Dragonborn will not necessarily have the ability to absorb the soul of a dragon.
Rather, in Skyrim, the player is told to kill dragons and absorb their souls to become a stronger Dragonborn. Along with a Dragonborn's natural affinity with the Thu'um, the act of absorbing a dragon's soul greatly accelerates the time it takes for the Dragonborn to learn to use their shouts. Arguably, the player would not have been asked to do so if they were not expected to go up against a potential 'end of days' power, nor if the dragons were not already flying around causing a great amount of chaos.
The player is able to find a friendly dragon, and will be exiled from particular groups, if they choose to kill him.
Who can Thu'um?
asthe Dragonborn you can shout."
As the Dragonborn it is easier to shout.
Not really an inaccuracy, but it is worth addressing that being Dragonborn did not give you the ability to shout. It just gave you an affinity to do so. As previously referenced, considerable study could lead to anyone being able to shout.
1 Note that lore expressly mentions the inherited blood of the dragon. To be Dragonborn, one must have both the blood and the soul of a dragon. Speculation aside, the future generations are still acknowledged as being Dragonborn.
Emperor Tiber septim was dragonborn and he was known to shout however he did not absorbe any dragon souls because most were extinct around that time,instead he learned from wulfharth. No other septim used the voice after him and likely were not "true" dragonborn in the sense that they have the blood and soul of a dragon they were more "alessian" dragonborn and not tiber/miraak/skyrim dragonborn. Blood passes down through family heritage, but not the soul.