I am using Windows 10, and the Skype App that came with it. (Not Skype for Business.)
I have its icon in the taskbar.
But when I get a message, the taskbar icon doesn't change to show me that I received a message.
I found a way to get it to show that a message was received.
(Note: This will work for future messages, but not retroactively.)
In the Start Menu, I right click on Skype > More > Turn Live Tile On.
Then on the next message you get, it will show that 1 message was received:
UPDATE: This does NOT always work. The notification count does not always appear. I recommend switching to Classic Skype (aka "Skype for Windows") if possible.
Turn on"Show badges on taskbar buttons". The icon still does not flash, but you get a number bubble of the number of new messages you have. Setting>Personalization>taskbar