How to delete matching brackets in VS Code?

There is an extension called Bracketeer that does what you want.

First, install with

Ctrl–p, then ext install pustelto.bracketeer, followed by an enter.

You can then add the following to your keybindings.json

  {
    "key": "ctrl+alt+backspace",
    "command": "bracketeer.removeBrackets"
  },

You can then use ctrl–alt–backspace to remove matching brackets.