Adding to the previous - I found a workaround (July 2022) to make a download / backup of Google Chat text that does not require Google Takeout.
I am not sure how long this workaround will last, since this method seems to rely on 'Google Hangouts' - which I understand is being deprecated.
To export / backup your Google Chat do this:
Go to the Gmail account that was used for the chat. Enter into the search bar: in:chats
This should bring up a list of chats you have participated in.
Click on the chat you wish to save or copy - it should open all of the threads of that chat in a list - (it looks similar to threaded /nested gmail conversation).
Click on the Print All icon (printer icon) top right of the gmail window, which should then open a print dialogue box.
For me, this allowed me to "cancel" the actual print dialogue box, and then I am left with a single html page showing the formatted entire chat thread, which I was then able to copy and paste into a new email or wherever.
If it works for you, give me an upvote :-)
Hope this method stays available as it's better than nothing.
I followed this article to discover this method...