I was having the same problem on both my phone AND computer. The problem started when I reset my phone to factory conditions and started over. For my computer, I solved it by uninstalling the extension and installing the App. For easier access, I right clicked on the app from Google Chrome Apps and told it to make shortcuts on my desktop and menu. The menu shortcut didn't work (I'm on Linux) but the desktop did. So, I just edited the .desktop file, copied the command, edited the menu entry and pasted the correct command. I have no idea why Google Chrome thought a different command was needed in the menu.
For the phone, I had previously cleared data without success. I went into Hangouts -> Settings and clicked on each account listed there, signing out of each one, until none of the accounts were signed into Hangouts.
I then searched for Hangouts in the Play Store, uninstalled the dialer (maybe not necessary) and uninstalled the Hangouts Updates (as suggested in the post just above and others). It's not possible to fully uninstall Hangouts. I reinstalled hangouts and opened, but it did not ask for any permissions. However, some phone numbers had names. From here, I archived those phone numbers without names. Then, for the first time ever, when I clicked the + to send another text and typed a name, it showed up! I sent a text, and the name showed up along with the full history of the conversation from before I archived it. I also reinstalled the dialer.