Hi,
Even though the symptoms appear to be the same that are not.
Technically, and somewhat semantically I know, they are on different devices running different apps (albeit with the same name) on a different OS.
having said that there are now three known fixes on the Mac version.
2 involve ending the Dock Process either in Activity Monitor or the Terminal App. The other involves tuning Off Notifications for the Messages app in the System Preferences > Notifications.
In the Mac version you set the option for Banners to none and turn Off every other option (Sound and Dock being the most demonstrative) whilst the Messages app is running.
Then Quit the Messages app.
Close System Preferences and restart the computer before Restarting the Messages app.
It would seem this would be the most likely thing to work on the iPhone.

8:00 pm Sunday; March 13, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad