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Many thanks for getting back to me,
Yes, you are absolutely right. Currently, even if you highlight everything before copying and pasting, including the name and date, it will only paste the message into other apps. However, as a work around I sincerely suggest hold mouse for selecting the message with Time Stamp and name of the person you want then use CTRL+C for copy while you are still holding mouse and use CTRL+V for paste into another app.
I have been testing your query from my side and based on my test result the above workaround allows me to copy the message with sender name and the timestamp. you can refer to the screenshot below:
Result: pated into word.
I look forward to your update. Please feel free to let me know how it goes.
Thank you for your precious time. Have a nice day.
Sincerely,
Libeamlak | Microsoft Community Moderator