Google Meet (web, in particular) has live captions
Is there any way to record (and save) this [whole/full] meeting transcript?
Yes, but not using Meet and the natural language processing it provides directly.
Even if you record a meeting, captions are not recorded and don't appear when you play the recording.
What you can do is record the meeting, download that recording, and then use some third party app to create a full transcript from that video. For example, you could upload to youtube, and then see the transcript based off of their processing.
There is also a chrome extension called Tactiq that can save a full meeting transcription. However, the processing Tactiq uses is not the same one Meet uses.