I had accidentally enabled a different mode for the tab key. Fixed it by pressing Cmd+Shift+M (for Mac), or Ctrl+M (for Windows).
From the Visual Studio Code Keybinding docs:
| Key | Command | Command id | | Ctrl + M | Toggle Use of Tab Key for Setting Focus | editor.action.toggleTabFocusMode |
The current tab control mode should also show up in the status bar:
