@Artyou Global Thank you for reaching out to Microsoft Q&A. I had checked with internal team and sharing the below.
Unfortunately, back up and downloading of chat threads are not supported in ACS. Communication Service chat thread and messages are retained until explicitly deleted. You can use the the Chat Thread Rest APIs to get/list/update/delete messages and save them in proper format and migrate them to another service.
your feedback is noted and I have relayed the feedback internally to our product engineering team.
We also have a feedback item on this, if you wish you may upvote on the Uservoice post - All of the feedback, you share in these forums will be monitored and reviewed by the Microsoft engineering teams responsible for building.