This is what I did and it seems to work well. You need to create a chain of unit upgrades including the unique units similar to the following:
Warrior => Jaguar Warrior
Jaguar Warrior => Swordsman
Swordsman => Legionary
Legionary => Immortal
Immortal => none
This way, everything will upgrade properly. The Aztec's Jaguar Warrior will upgrade a Swordsman. Other civs that only have regular Warriors will also upgrade to Swordsman because when the game is trying to upgrade it first looks sees Jaguar Warrior. Since these civs cannot build this unit, it looks to the next unit which is the Swordsman.
In this example, units would upgrade like this:
Aztec Jaguar Warriors upgrade to Swordsmen
Roman Warriors upgrade to Legionaries
Persian Warriors upgrade to Immortals
For all other civs, standard Warriors upgrade to standard Swordsmen.
You can do this same thing with all the other units. Just group them into categories, such as defenders, transports, etc., and create a upgrade chain using all the units of this type including the unique units.
This is how I have my other units set up:
(NOTE - Not all units are listed, and a couple of the units listed are new)
Scout => Explorer
Explorer => none
Archer => Bowman
Bowman => Longbowman
Longbowman => none
Spearman => Impi
Impi => Pikeman
Pikeman => Hoplite
Hoplite => Musketman
Musketman => Musketeer
Musketeer => Rifleman
Rifleman => Infantry
Infantry => Mech. Infantry
Mech. Infantry => none
Chariot => War Chariot
War Chariot => Horseman
Horseman => Mtd. Warrior
Mtd. Warrior => War Elephant
War Elephant => Knight
Knight => Rider
Rider => Cavalry
Cavalry => Cossack
Cossack => none
Tank => Panzer
Panzer => Modern Armor
Modern Armor => none
Catapult => Cannon
Cannon => Artillery
Artillery => Radar Artillery
Radar Artillery => none
Coracle => Galley
Galley => Caravel
Caravel => Galleon
Galleon => VOC Galleon
VOC Galleon => Transport
Transport => none
Frigate => Man-o-war
Man-o-war => none
Fighter => Jet Fighter
Jet Fighter => F-15
F-15 => none