I dont think OP or me have any problems with asymmetrical civs or having unique units in the game. It is just about slapping on names on previously perfectly fine cards and/or units. Which is also fine, whatever, it is a neutral change. It neither degrades nor improves anything about the game. I feel either 2 name convention or a toggle in game settings will definitely work the best.
History reference name + [ Cavalry Combat ]
History reference name + [ Fencing School ] - names like this would work for everyone.
And a toggle in game settings to show either the new historic name or the legacy traditional name.
No one is asking for making civs or units symmetric. Or naming every unit Hand Cavalry1, Musket Gunner2, 
That is absurd.
The original devs had a game design in mind. Some things were abstracted, some details omitted. I remember the original campaigns, I never bothered me when units were reused for different civs. I had enough suspension of disbelief and imagination, that it didn’t matter to me.
In the heat of battle, no one is thinking I should send the [long historical reference name]. People just use cav combat, infantry attack, artillery hp.
The thing is, the historical reference name used to mean something. They were reserved for cards which were special and had a special effect.