Do I need to use an extension to access Google Meet recordings

No, you do not. The extension is just for convenience.

What makes this different from all the others?

We use the new Google Meet API's, this means you will be able to use this when you start a Recording or Transcription from within Google Meet.

Where is my data stored?

Google Servers hold the master copy of your data. Meeting Playback uses the Google API to retrieve data when you request it.