Universe in Brief
Mutants do exist in this world. However, they are not as numerous as it seemed to be portrayed in the X-Men movies.
There is a super-hero group that the entire world knows about, The Guardians. Its members include Celestia & Citadel (and some others that are usually not noted in the media). They are the heroes the world depends on.
In order to lower the fear that the possible existence of powered people may cause, the world created the organization called Stronghold. They are a public organization. They do have technology that has been proven to be able to subdue powered people, especially powered villains.
The current universe is loosely based in the Marvel Movie Universe. As such Marvel Comics and their movies do not exist in the universe, but the DC ones do.
Spider-Man does exist in New York. Most of his activities have been kept from the general public via the efforts of SHIELD. Only Spider-Man one and two have occurred.
The Strategic Homeland Intervention, Enforcement and Logistics Division (SHIELD) does exist and is highly classified. They have been able to keep the actions of most meta-humans from affecting the general public.
As far as the X-Men are concerned, the first movie did occur. The registration act was not a public act, but one conceived in ‘secret sessions’ of congress. Xavier’s School for Gifted Youngsters does exist, but there is no public knowledge about its true purpose. The second movie did occur as well, except that the event toward the end where Xavier was finding all the humans or mutants did not occur (they got to him before that happened). The X-men do exist. However, their exploits are kept well in secret by either their own efforts, or the efforts of SHEILD.
It is only recently that the events in the first Fantastic Four movie have occurred with Reed Richards et al have returned from the Von Doom Space Station with Meta-Human abilities.