And if both lieges have the same authority, and the vassal conquers land that is not de-jure either liege's realm, who gets the tax and levies from that land? Does the vassal rule that land as effectively independent, or vassal to a third realm? What if a vassal holds lands within the liege's realm and beyond the liege's de-jure land, is that land beyond those borders still part of the liege's realm?
To me, the easiest and simplest way for the system to work would be to have a primary and secondary liege. Your primary liege is the first lord you swore fealty to or was your liege at the start of the game. For example, lets say you are the Duke of Barcelona in 1066 and there is some Muslim consolidation in the early game so you decide to become a vassal of France, who is now your primary liege. You then inherit the Duchy of Bavaria so the Holy Roman Emperor is now your secondary liege. Any land your control in the realm of the HRE pays it's obligations to the Emperor, including non-de jure HRE territory that is already part of the HRE, like Provence for example. Any other territory you control pays its obligations to the King of France. If you become a king, de jure HRE or not, you would only owe obligations to the King of France for territory that is de jure France, if you had any. If your new kingdom was de jure HRE then the Emperor gets all of your feudal obligations. If, for example, you become King of Croatia, the only obligations you would owe to France or the HRE would be related to de jure territory, everything else would be considered part of Croatia now.
As for tertiary lieges, I wouldn't go beyond two lieges for simplicity's sake. If you would inherit territory that would give you a third liege it can either go to another dynasty member or just revert the king/emperor. You could then get claims that would allow you conquer the territory which would then be governed by the rules above. Anyone who already has two lieges are ineligible for claimant factions unless the title in question is already part of their lieges' domains.
What exactly dictates when a vassal should have multiple lieges?
I would say only inheritance or being installed by factions. If you conquered something, it's yours. If you are a king and inherit lands in another kingdom you only owe de jure obligations if they had high enough crown authority, level three, which normally stops lands from being inherited outside for the realm. If you are a vassal and gain a title equal to your liege you only obligations on de jure territory as well. If all of your domains are outside their de jure territory then you are completely independent.
Does a vassal become independent from a liege on losing all titles that were counted as part of that liege's realm, even if the vassal still holds land elsewhere?
Yes, unless that liege is also their primary liege. If you want true independence you would have to fight for it.
If a duke controls lands between two kingdoms, and one kingdom declares a claim war on that duchy, how is the vassalage assigned?
Could you add a little more detail. Like, is the duke is independent?
If a vassal can be vassal to multiple lieges, can a vassal be independent and vassal at the same time, and how does that interact with war targets?
Yes and no. Using William and Normandy as an example. William wouldn't strictly speaking, be the King of France's vassal but he would owe his feudal obligations, taxes and levies, to France. So, the King of France can't revoke his titles but neither can William have a council position and so on. William would owe increased obligations under higher crown laws but the actual feudal contract itself can't be changed. The Kingdom of France could still declare a de jure war in order to stop de jure drift. If someone wanted Normandy they would declare against William since the lands are currently part of his domains. If you are an independent duke and inherit titles in a kingdom then they would stay independent like the current game rules work.
Can a vassal declare a war for partial independence?
No opinion really. I feel like this is something that would need to be playtested to see what works best.
And how does all this apply to other parts of the world?
Given the fact the game treats the rest of the world as Feudal Europe I don't see why it couldn't do the same for shared vassalage. I would limit it to feudal rulers though and not make it available to Clan and Tribal Rulers.