Note: Figured out the issue. You have to select the text and then click on what you selected for the chat bubble to come up, then press 'Copy'. Leaving this up for others to see.
--------I want to be able to select a feature, copy a field value from the feature's pop-up and then add a new feature to a different layer and copy that value into a field of the new feature. Can this not be done? It seems that you can copy-paste text when you are editing a field, but you cannot copy straight from a pop-up.