You can actually get the same behavior for received messages. This only happens when there are multiple replies from the same author in the same thread/chat. And I'm OK with it as it minimizes the amount of screen estate it takes for the messages to be displayed. It's an inconvenience when you want to copy just a word or two, sure, but the tradeoff is not worth it IMO.
The triple left click still doesn't make it apparent that you have selected the entire text because the options are still displayed. The message text should simply be expanded with the three option not hiding any part of the text.
You'll get no arguments from me about that. It's unfortunately the current state with the ellipses in place, but hopefully Microsoft will fix this issue.
What I have proposed is (what I consider) a simple work around to be able to grab the text in the meantime.
Let's wait and see!
P.S. When you triple click and select all the text (and copy), move your mouse away from the chat bubble and it should show you that you have selected the text. Might help to see if you have successfully copied the whole text.