Dear Bradley Morris27,
Good day.
After thorough research, testing from my end and consulting, it’s been concluded that currently copying and pasting from a Microsoft new Teams chat window no longer includes the sender's name and timestamp. You need to use the workaround like you suggest in your description , " I have used is to highlight your text and while holding down the left mouse button (Do not release after highlighting) press CTRL+C".
We have find similar case in our feedback portal and some users already report the issue Teams chat. copy & paste · Community (microsoft.com). Kindly upvote their request.
We apologize if we are not able to meet your requirements; however, thank you for bringing this to our attention, we will surely raise you concerns to our Product Team and hopefully this can be included in the future updates.
In the meantime, please feel free to post this in our Feedback community by visiting Microsoft Teams · Community as it is a community monitored by our Product team to help improve user experience with Teams, via feedback given by our valuable customers.
For other concerns, please do not hesitate to add you post in the Microsoft Community Team.
Thank you for choosing Microsoft as your partner in business.
Sincerely
Eben Ezer Tres | Microsoft Community Moderator