It was indeed on the Manchester COA. But never originally.
It was awarded to the city of Manchester in 1958 because of the city’s origin links to Ancient Rome, and added to the mantling on the COA for a short period of time.
Just like they have the name ‘Manchester’ despite being in Trafford, they adopted that new feature from the Mcr COA just after the Munich Air Disaster and wore it on their 1958 FA Cup final kit because it was awarded to the city almost exactly at that time.
It soon disappeared from the Mcr COA (modern one here):
View attachment 89996
But will always be associated with United in 1958. As seen as their fan groups have retaken it since we stopped using it.