To be honest, addition of Tamar got me hyped originally as a native, but after closer examination, things turned different. I'm aware of popular meme that was on this forum and I would expect that a civilization that literally got memed into the game would most likely get a meme treatment as a result. Still, that was kind of disappointing. And due to this I dont really expect that there will be actual relevant changes. Nevertheless, here are my two cents on Improving Georgia.
First things first, people already mentioned that one of the main issues with Georgia is a religious civ with no bonus on starting religion, essentially that would leave Georgia a crippled mess. Second is the very fact that it is purely religious civilization, which, imho, is a mistake. Considering that large part of our history is that of bloodbath and misery against numerically much more superior foe and despite this we have for some odd reason survived to this very day, and considering that large parts of our traditions, be it national dress to things as mundane and simple as food is delivered from military legacy (our foods were essentially made to fit soldier's life) I think that it should be Religious-Military, with strong defensive bonuses that goes beyond a unique building that nobody cares about. Now, I admit that some suggestions that I may propose here are not really original. In fact, for the most part I was kind of inspired by newly added Hungary, as well as maybe getting Ideas from Russia and Japan, of all things. I also admit that some suggestions here may be seem OP, but again, after seeing Raven King's special abilities, I've decided to give it a shot. Though I doubt that my suggestions will make any notable difference, I'll still give it a try.
What should be said from a get go is that Civilizational and Leader abilities are...swapped. Client state swarm was a common strategy in Georgian politics both before and after Tamar. It may be dated back to Pharsmanes I of Iberia which made Armenia a client state with Rome's aid and installing his brother as a king there. And it was used even during the reign of Heraclius II. So I would say that if there is a mechanic regarding city-states and clients, it should be civilizational trait. meanwhile Golden Era is associated specifically with Tamar, so that should be her bonus instead, if there will be such. Now as for specific suggestions:
Civ Ability -
Strength in Unity: Damaged units face no penalties when fighting on friendly (Native, Allied) territory or from two tiles from Empire's border. Allying with a city-city state that shares its religion with Georgia will make it a Client State rather than average ally, which will bar other civilizations from sending envoys to that city state and give Georgia ability to levy city's units for free for indefinite time.
Explanation - This will make Georgia an actually solid defensive civilization that would turn every invasion from enemies side into a horrid meatgrinder as it historically was and a good ally to have on your side. The part about city states will make Georgian relations with same-faith city states far more rewarding, as spreading religion is already a difficult matter, while some civs and leaders like Corvinus get "double envoy" bonus without such demanding pre-requests. Now spreading religion is basically part of your literal expansion, as it potentially turns City-States an autonomous part of your Empire in all but name. This will also make other players to use religion more extensively if Georgia is present in game, to prevent permanent loss of potential city state allies. So in global game perspective, this addition actually makes religion even more strategic and important factor. To balance the ability if it seems too OP, levied army's upkeep can be made to be depended on Georgia itself with +25% additional cost for maintenance. Also, it's can be seen as kind of historically justified, as we lost our hold on North Caucasian vassals only after they converted to Islam after invasions by Golden Horde and Tamerlane (though we also had muslim protectorates to the south and east so there's that).
Leader Ability -
Glory of the World, Kingdom and Faith: Dedications chosen at the beginning of a Golden Age or Heroic Age also grant their respective Normal Age bonuses towards improving Era Score, in addition to their regular bonuses. Golden age after three consecutive Golden Ages turns into a Heroic Age. +1 Prophet points for every city-state and civilization encountered.
Explanation - As I have mentioned above, I find Golden Age bonuses quite fitting for her, so I would resign it as leader trait mostly unchanged barring the last part about the heroic age. Also a prophet points boon will make sure that Georgia has a good chance of getting a religion.
Unique Unit -
Aznauri: Heavy Cavalry/Cataphract, +4 combat ability for each adjacent enemy unit.
Explanation - Most of our battles were won despite huge numerical disadvantage, be it Battle of Didgori, Shamkori, Basiani and so on. I think that above mentioned bonus would represent such encounters more or less accurate. Combined with national trait, invading medieval era Georgia (or its friends) would be... messy, to say the least. Also, if you thought about it, yes, its an inverted, hellish parody of the Black Army. I'm quite original, arent I?
NB - I am also not fond of Khevsurs as the unity simply represents very specific part of the country, not the state in general as Aznauri would.
Unique Building -
Tsikhe: An Encampment. Double the damage against enemy units than average Encampment. +20% production bonus towards units when at war.
Explanation - Tsikhe are Forts and Castles, which I think can be more accurately represented as Encampments rather than city walls. Its bonus damage against enemies I think fits well with defensive play. +20% bonus during war will allow rapid mobilization of forces to counter the invading enemy.
Unique Building 2 -
Georgian Monastery: May be built on territory of Allied Civilization (Maximum 3 in each allied civilization). Each Monastery grants +1 science and +1 culture to Georgia. Ally, on whose territory the monastery is built will gain +10 faith points. Ally gets +5 opinion for each monastery present on their lands (Max. +15)
Explanation - Medieval Georgia was known for its oddly large amount of spiritual centers present in foreign lands such as
Byzantium, Bulgaria and Levant. These centers were vital for cultural and scientific advancement of the nation, as vast volumes of different works were translated into Georgian by the monks such as George the Hagiorite, Euthymus the Athonite, John the Hagiorite and so on. +1 science and culture will represent the vital influence of these cultural centers. It must also be noted that high number of monks there were usually distinguished politicians and military officers in early life such as Gregory Pakourianos, who founded Bachkovo Monastery in Bulgaria, but most important perhaps was Tornike Eristavi, who, at the request of the Byzantine Emperor, played large role in convincing David III to aid Constantinople in suppression of the revolt of Bardas Skleros. Tornike Personally led the army against the rebels. I intended +5 opinion relations bonus to be represent such occasions as well as general benefits of cultural exchange that happened between the civilizations.
NB - To be honest, I also came up with this because of my personal opinions. I am generally fond of concepts of buildings that benefit both the builder and the ally, as well as forming strong and everlasting alliances.