Should you still have some issues, please follow below:
To be able to copy Teams message without timestamp and sender information, all you have to do is to NOT use double click of mouse to select the text you want to copy, neither Ctrl+A.
Just use mouse cursor to select a text you wish to copy, once highlighted, press Ctrl+C and paste with standard Ctrl+V.
This is incorrect, you have to carefully and deliberately drag from the start of the text. If you ever start from the end of the text when highlighting you will highlight an invisible timestamp. Additionally if you ever highlight two chat messages the timestamps will appear no matter what direction you start highlighting from. Why the timestamps can be highlighted invisibly to be copied makes no sense since you can highlight them to copy the same thing anyway. Why doesn't Microsoft just remove the invisible timestamps and leave just the visible ones?