The Forgotten Realms is one of many D & D worlds, but this is the default in any edition.
It's a common misconception that Forgotten Realms is the default setting for D & D. In fact, the "main" or "default" world varies depending on the game's edition, and none of the these do not mention the Forgotten Realms.
The closest is the 5th edition of D & D, where each D & D universe is officially considered part of a multiverse of all D & D worlds, which means that the Forgotten Realms are a reality. part from the D & D universe, but no more than, for example, Eberron or Greyhawk. To quote Jeremy Crawford:
Does the table RPG #dnd have an official parameter? The answer is yes. This setting is the multiverse, which includes all #dnd worlds.
What was the main frame?
The 1st edition of AD & D had no default setting and did not make a clear distinction between World of Greyhawk and what was "generic". When introducing the Forgotten Realms, many central elements or Greyhawk, such as Underdark and Great Wheel cosmology, were simply included in the Kingdoms.
The 2nd edition of AD & D did not focus on one basic parameter, but on several parameters. The sourcesbooks of this edition often included sidebars allowing adaptation to one of the many parameters introduced in this edition of the game.
The 3rd edition of D & D has made the World of Greyhawk the default setting. We see this with the deities in the Player's Handbook.
The 4th edition of D & D detailed the Points of Light or Nentir Vale parameter in the Dungeon Guide, and is considered as its fundamental parameter.
The 5th edition of D & D includes many references to the Forgotten Realms, but if you read the Player's Manual, you will also find references to Greyhawk, Dragonlance, etc. See my answer to Forgotten Realms is the default setting in 5th? for my previous description of this.
In what world does each game live?
The Forgotten Realms is a continuous and consistent parameter. If you play a game from the 5th edition of D & D, you live canonically in a world in which the cataclysms of AD & D 2nd, D & D 3rd and D & D 4th have occurred in the past.
however, the DM is completely free to change all the details about this, which is in fact almost inevitable, because very few DM have read each collection of sources and each novel of Forgotten Realms, their understanding of the world will therefore naturally divergent. This is the normal state of affairs, according to D & D 5th DMG p.4:
The world in which you define your campaign is one of countless worlds that make up D & D. multiverse, a vast range of planes and worlds where adventures take place. Even if you use an established world such as Forgotten Realms, your campaign unfolds in a kind of mirror universe of the official set in which novels, gaming products and digital games from Forgotten Realms are supposed to unfold. The world is up to you to change and change as you explore the consequences of player actions.
In other words, if my D & D group gets drunk and burns Candlekeep, this event is only true in my campaign. As an extension of your campaign, Candlekeep could be doing well. That's what we mean by "mirror universe". If you are performing a game in the Forgotten Realms, it is your group's own interpretation of Forgotten Realms.
Related to this topic, see Jackson Crawford's video When Myth Is Inconsistent.