annoyance: latest version of Skype under Standard (non-admin) user on macOS 10.13
So, I was right that Skype has a problem, because it is prompting me to perform the auto-update, but supplying the credentials is not enough.  I have to kind of "hand-hold" Skype through the update process, and open the correct window even though there is no indication from Skype that I need to do this.  I was still having this problem after upgrading to Mojave (10.14) and I'm glad I finally figured out this work-around.



In order to successfully update Skype in macOS as a limited user, you must do it "manually".

When the pop up notification appears saying An update is ready to install. Skype is trying to add a new helper tool. and asking for authentication, do not enter your admin credentials yet.

First, click on the Skype menu then About Skype.

With the About Skype window open, now you can enter your admin credentials.

After you've entered your admin credentials, you should notice a green Update button is now active in the About Skype window.  Click the button.

Sometimes I have to enter my admin credentials again when the update process starts.  This is normal.

Skype should automatically reload with the updated version.  Check the About Skype window to confirm you are on the latest version.