The multiplayer is centred mostly around progression and customization. Two players can have widely different armies but they have the option to have anything that their opponent has. There are 3 ways to customize an army:

• Retainers
  
• Veteran skill upgrades
  
• General's skill tree

Retainers affect the stats of the entire army. The general's skill tree affects the stats of specific unit types or the entire army or just the general's abilities (which may also affect other units). The veteran skill upgrades only upgrade the stats of the veterans.
You unlock retainers by taking over territories on the campaign map which you do by winning. You level up your general and unlock new skills based on the experience your avatar has. Veteran units can be added after a battle and they gain more xp based on the amount of fighting they do in battles. That's basically it. In the next sections I'll talk about the problems.

For some brilliant reason certain veteran upgrades don't do anything. This problem usually occurs with reload upgrades. It's important that you don't waste your points on these upgrades because you can' t just reset veterans. You will have to start a new one and level it up all over again. Others have already tested the effects of upgrades so I'll just link to their guides here:
https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=122987621
https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?l=german&id=644200596

Some veteran units are veteran only units. You cannot delete them. This means that if you want to reset their upgrades to something that you think is better you can't. Some of these units are crucial in battle so it is very important not to screw up with the upgrades. There's a mod however that fixes this problem:
https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=1292326986

This section is not relevant at the moment since the clan servers are down. I will talk about that in the next section. However if they ever get fixed again the following will be relevant:

Originally some veteran upgrades were allowed based on whether you are in a Shogunate or in an Imperial clan. Certain units are tagged as shogunate and others imperial and you could only get the level 9 upgrade on them if you are in a clan of that type. Now when you switched clans the game would wipe out all your level 9 upgrades. This is pretty bad since it gives you extra work for no reason.
The way it works is that the game only punishes you when you switch to a clan that is not the one that your level 9 upgrades belong to. So there are two ways to solve this problem:

1. Have a separate avatar where you switch clans. This avatar doesn't need to have anything in it. Then the 'punishment' only applies to this dummy avatar. Then you can switch back to your other one and upgrade units. This way you can have both shogunate and imperial level 9 units in the same avatar. Now bringing both types of units in a battle is considered a cheat and will get you banned from tournaments. However storing the units in the same avatar is fine.
   
2. Although I haven't tested this there might a smarter way of doing this. You can have a separate Shogunate avatar and an Imperial avatar. As long as you make sure that when you switch clan you switch to the clan that is compatible with your units in the avatar you are currently in you should be fine. This helps avoid cluttering the veteran panel of your avatars and gives you more space.

Level 9 veteran skill upgrades require you to be in a clan of the appropriate type and to have clan tokens with which you buy that upgrade. The clan servers have been down for nearly a year now and you can no longer get clan tokens which you used to be able to get from playing matchmade battles. I have made the following fix to this:
https://psteamcommunity.yuanyoumao.com/app/34330/discussions/0/508449490226688476/

Finding players on Shogun 2 in general has become more difficult for 2 main reasons:

• The leaderboard no longer works which means there's less incentive for people to play matchmade battles.
  
• CA removed the chat so it's harder to communicate.

There's less chance that you'll bump into people which is why it's best to join online communities. I will link the most active I know where tournaments are hosted every now and then:
RepublicArena (doesn't only focus on Shogun 2 but it's still the best place):
https://discord.gg/SDU8yuxGeu