False. Knowledge of one being Dragonborn is limited to Guards, dragon death event and quest specific dialogue. Kill command a dragon and everyone in the shared group will run and say "You, You're Dragonborn." and things like that, even if you already captured the soul and it doesn't reproduce one. Guards are aware when you shout using the Misc dialogue function and a handful of NPCs during quests like BallGroove the greater. The world of Skyrim is very static to the events that unfold. In Oblivion Every NPC in all the 'cities' would have custom dialogue for closing the gate out of the city walls. You were recognized globally as the hero of Kvatch and the Grand Champion for the arena. The landscape would change slightly. In Morrowind every NPC is aware of your status at the end, the world changes noticeably. As the disposition system was removed from Skyrim your status only changes for people and factions you directly helped. There is no global awareness in Skyrim like the predecessors.
Actually, number 1, there IS still a disposition system in Skyrim. It's just incredibly dumbed down. Every time you do a mission for someone you can raise or lower their disposition by 1 or 2 points. See here as a good example of where a quest raises or lowers disposition with 2 different people:
http://www.uesp.net/wiki/Skyrim:The_Taste_of_Death It's incredibly dumbed down compared to Morrowind or Oblivion but it is actually there.
Number 2, the word "everyone" was a hyperbole, I'm well aware that not everyone acknowledges the fact you're the hero. It isn't like Morrowind where after you defeat Dagoth Ur literally every NPC will say "Oh gods, you're the Nerevarine! I can't believe you're talking to me of all people, I'm so honored!" But you're still recognized as the Dragonborn every time you do something, such as shout, kill a dragon, and people will mention it a lot during quests. You're still seen as a hero at least part of the time, while in Fallout 2 no one recognizes you as a hero, and in fact you're met with discrimination in a lot of places because of your tribal status.. Same for Morrowind, before you're recognized as the Nerevarine you're met with a lot of discrimination because you're an outsider, a greenhorn. No one respects you until you do things for them.
Meanwhile in Skyrim, because you're the Dragonborn, a lot of people praise you for it for no reason, even if they don't know you. The point I'm trying to make here is that in Morrowind and Fallout 2 you start off as a literal nobody, and no one gives a guar shit about you or who you are. You have to earn their respect. Meanwhile in Skyrim you kill one dragon, which any random guard can do with a bow and arrow, and everyone will gather around and shout "OMG THE DRAGONBORN CAN I HAVE YOUR AUTOGRAPH?!"