Is The Elder Scrolls V: Skyrim the only Elder Scrolls game where you are Dragonborn, and if so, are there other Dragonborns in the other games?
Given this seems to be more of a lore question, it is good to make note that elderscrolls lore is complex in detail yet leaves some up to the players imagination and interpretation. I recommend reading the wiki and other places on the internet for interpretation on lore.
The only player character confirmed to be a dragonborn is in Skyrim, all others are up to interpretation with varying degrees of support as to whether or not they are a dragonborn.
I will cover the facts, though this is from memory of personal exploration and reading, I could have missed important facts in some places. I will edit if necessary. (And I did more reading, esp. on the dragonborns before Saint Alessia. Saint Alessia was the one to make a contract with Akatosh and have her bloodline infused with dragon blood, making them dragonborns)
First, Paarthunax states that dragonborns are mortals blessed with the Dragon Blood of Akatosh, the Father of dragons. Also note that some may be dragonborns without notable or recorded use of the thu`um.
To cover the known facts though, Skyrim is the only elderscrolls game where the player is confirmed to play as the dragonborn. Definitely the only one so far with the ability to use the Thu`um.
In Oblivion, the player character is not able to wear the amulet of the kings, which in lore can only be worn by those of the emperor's bloodline who share the blood of Akatosh. Note, that it is stated in Skyrim that the emperor's bloodline, the descendants of Talos (but actually Saint Alessia), are all dragonborns by blood. I am currently uncertain if the amulet requires the wearer to only have the blood of dragons or to also have the blood of Alessia. This apparently is up to interpretation, but most likely, since the amulet was used in the forging of the pact between Saint Alessia and Akatosh, and it is noted to contain the souls of the emperors, starting with Saint Alessia, I believe it requires the wearer both to be dragonborn and a descendant of Saint Alessia, which honors the origins of the pact.
I originally thought that being unable to wear the amulet of kings meant that the player is not dragonborn, but due to the necessity of Saint Alessia's bloodline for the amulet to be worn, as I prefer to believe, means the player could be a dragonborn, but must not be of her blood if so (An example of a dragonborn not of Saint Alessia's descendants is obviously Miraak, the first dragonborn). So the player character in Oblivion is also up to interpretation on whether or not they are a dragonborn.
The only emperors i know not to be dragonborns are the current one in Skyrim, who is related to a Dark Brotherhood quest. And the underking from arena or Daggerfall, i cannot remember which. The underking tricked people into thinking he was Uriel Septim, the one who died in the beginning of Oblivion.
Arena, Daggerfall,and Morrowind all are up to interpretation. The Nerevarine (player character in Morrowind) is mentioned briefly to be a dragonborn, but there is speculation that this means they are under the rule of the Imperial Empire, given they were sent here by Uriel Septim. I suspect the player character in the game Redguard, where the player character actually has a defined history and personality, is not related to
Talos Saint Alessia nor has any mention of Dragons blood (as far as I know), thus can be confirmed as not a dragonborn. No idea about battle spire, but I doubt the player character is a dragonborn.
For more information, I recommend doing some wiki reading: Unofficial Elderscrolls Pages on Lore of the Dragon born
As for non-player characters, I agree with prijatelj that Uriel and Martin Septim are most likely dragonborn. Additionally there is evidence to suggest that the player character from Morrowind is dragonborn as well, though this is highly subject to interpretation.
During Morrowind's main quest, the player collects a information regarding the Nerevarine, a prophesied hero of the dark elves, whose appearance would be signaled by a number of qualifying factors. Throughout the main quest, your character inevitably fulfills most (if not all) of the qualifying factors, convincing most NPCs that you are, in fact, the Nerevarine. The following is a passage from one of those prophecies, known as "The Lost Prophecy" (emphasis mine):
From seventh sign of eleventh generation,
Neither Hound nor Guar, nor Seed nor Harrow,
But Dragon-born and far-star-marked,
Outlander Incarnate beneath Red Mountain,
Blessed Guest counters seven curses,
Star-blessed hand wields thrice-cursed blade,
To reap the harvest of the unmourned house.
However as a prophecy, this passage is inherently subject to interpretation.