Superhuman Registration Act
The SRA does not exist in the MCU, most likely because of the number of "superhumans" in-universe at the moment. The Avengers are currently the only real organisation in the MCU that are in the eye of the media. They have no secret identities so to speak of, aside from:
Peter Parker (Spider-Man) and Matthew Murdock (Daredevil)
For more info on secret identities in the MCU please see: Does the general public know who superheroes really are in the Marvel Cinematic Universe?
The Sokovia Accords (the MCU's SRA equivalent)
The Sokovia Accords are the focus of the MCU Civil War. The Sokovia Accords are a set of legal documents that put the Avengers under the control of the United Nations (signed by 117 countries).
By signing the Accords the Avengers would surrender their right to choose where and when they intervene in situations across the world. If they do not sign the Accords, then the UN will attempt to neuter the Avengers. (Could they really...?)
The Sokovia Accords are put into place after the events of:
- The Chitauri invasion of New York (The Avengers)
- Malekith's attack on Greenwich, London (Thor: The Dark World)
- SHIELD/Hydra Helicarriers crashing in Washington, DC (CA: The Winter Soldier)
- Hulk vs Hulkbuster (Avengers: Age of Ultron)
- Ultron's destruction of Sokovia (Avengers: Age of Ultron)
Lagos (CA: Civil War)
All of the above events have killed or injured a number of civilians, and the Avengers have not been held accountable for these damages.
Not to mention the boom of catastrophic events since the Avengers have come into play. There are examples in almost every MCU movie to date of stuff going wrong, because of the actions of "superheroes":
- Thor - Puente Antiguo was leveled by Thor's fight with the Destroyer.
- Iron Man 2 - Ivan Vanko devastated the Monaco Grand Prix, and caused large collateral damage at the Stark Expo.
- Iron Man 3 - The introduction of AIM and the Extremis virus led to various civilian casualties and collateral damage.
- Ant-Man - The total destruction of a building, various collateral damage, and the almost-sale of a weapon of mass destruction (the Yellowjacket suit) to parties including Hydra.
What's the conflict then?
This causes a "Civil War" because half the team feel that they need oversight, they need to be held in check. As General Thaddeus "Thunderbolt" Ross says:
In the past 4 years, you've operated with unlimited power and no supervision. That's an arrangement the governments of the world can no longer tolerate.
The other side of this coin, as Steve Rogers says, is that: The UN is run by people, people have agendas, and agendas change over time.
Who's right then?
Tony Stark: We need to be put in check. Whatever form that takes, I'm game.
The crux of the MCU Civil War is that Team Tony believe they need someone to be responsible for them, someone to take the heat when things go wrong. Team Cap believe that they should be responsible for their own actions, and not be tied down to a group of people that, for all intents and purposes, could be motivated to divert/misuse the Avengers for their own ends.
Team Tony believe that having a "boss" for the team to answer to will not just legitimize the Avengers as an organisation, but absolve them of the blame that currently falls at their feet. In Civil War we see:
A woman attends Tony's MIT talk and accuses him of murdering her boy, who was working relief in Sokovia when Ultron decimated the town.
I believe this is a primary driving force for Tony Stark, and most of Team Tony.
Team Cap believe that they cannot trust an organisation to oversee them. Specifically, Cap believes that by signing the Accords, they would become lapdogs of an organisation. What if the Avengers need to be somewhere, but the UN says no because a small-time dictator doesn't want international interference with their regime? What if the Avengers are instead sent to this dictator's country, but with the mission of putting down the rebels/freedom fighters?