The Forgotten Realms is one of many D&D worlds, but it is not the default in any edition.
It's a common misconception that Forgotten Realms is the default setting for D&D. In fact, the "core" or "default" world varies by game edition, and in none of those is it stated to be the Forgotten Realms.
The closest is D&D 5th edition, where every single D&D world is officially considered to be part of a multiverse of all D&D worlds, meaning that the Forgotten Realms is part of the D&D universe, but no more so than, say, Eberron or Greyhawk. To quote Jeremy Crawford:
Does the #dnd tabletop RPG have one official setting? The answer is yes. That setting is the multiverse, which includes all #dnd worlds.
However, the Forgotten Realms is certainly given the most attention of any world in D&D 5e. It's the focus of the Adventurer's League organized play, as well as many official adventure modules. The ethnic groups of Faerûn are also the only ones described in the human race description in the Player's Handbook. This can give the impression that the Forgotten Realms is the "core" setting, but that's not strictly the case.
What was the core setting?
AD&D 1st edition had no default setting per se, and made no clear distinction between what was World of Greyhawk and what was "generic". When Forgotten Realms was introduced, many core or Greyhawk elements such as the Underdark and Great Wheel cosmology were simply included in the Realms.
AD&D 2nd edition did not focus on any one core setting, but had many settings. Sourcebooks for that edition often included sidebars for adaption to any of the many settings introduced in this edition of the game.
D&D 3rd edition made the World of Greyhawk the implied core setting. We see this with the deities in Player's Handbook.
D&D 4th edition detailed the Points of Light or Nentir Vale setting in the Dungeon Master's Guide, and that is considered its core setting.
D&D 5th edition includes numerous references to the Forgotten Realms, but if you read the Player's Handbook, there are also equal references to Greyhawk, Dragonlance, and so on. See my answer to Is Forgotten Realms the default setting in 5e? for my previous description of this.
Which world does each game inhabit?
The Forgotten Realms is a continuous and consistent setting. If you play a D&D 5th edition game, you are canonically inhabiting a world in which the cataclysms of AD&D 2e, D&D 3e and D&D 4e happened in the past.
However, the DM is entirely free to change any details of this, and in fact this is almost inevitable, since very few DMs have read every single Forgotten Realms sourcebook and novel, so their understanding of the world will naturally diverge. This is the normal state of affairs, according to the D&D 5e DMG p.4:
The world where you set your campaign is one of countless worlds that make up the D&D multiverse, a vast array of planes and worlds where adventures happen. Even if you're using an established world such as the Forgotten Realms, your campaign takes place in a sort of mirror universe of the official setting where the Forgotten Realms novels, game products, and digital games are assumed to take place. The world is yours to change as you see fit and yours to modify as you explore the consequences of the players' actions.
In other words, if my D&D group gets drunk and burns down Candlekeep, that event is only true within my campaign. In your campaign's continuity, Candlekeep may be fine and well. This is what is meant by "mirror universe". If you run a game set in the Forgotten Realms, it is your group's own interpretation of the Forgotten Realms.
Related to this topic, see Jackson Crawford's video When Myth Is Inconsistent.
