I'll contribute with a list of suggestions that I think it can improve the immersion, some of them might be already said by other people or other posts of mine though, and some of them might be too over the top/difficult to make. This list is subject to change or to be expanded in a near future.
Tournaments:
-A town with a tournament becomes more festive, the peasents are aware of the tournament, towns are decorated, etc
-Travelling competitors: Recurring competitors are indeed a thing in this game, it was in warband too. My idea is to expand on them, have them have a personality, you could find them in taverns along his friends or around people admiring them in town. Maybe have one have a special quest, make them create rivalries with the player or other competitors.
-Being recognized: After winning a large amount of tournaments you would be recognized by the commoners and lords, you would win a renown bonus with ladies or marriable characters.
-Bring back dedicating tournaments, even allowing you to dedicate it to lords and kings
-Increase difficulty, give reward to 2nd and 3rd place to balance it. The final fight is against a great warrior you would need to fight well to beat him.
Lord's keep:
-Populate keeps: Bring stewards, lord's children, counselors, high level soldiers who are in your party would rest inside the keep, have drinks, talk with them about current events and bring morale based dialog that would give you an idea on how to improve your party, like a lack of food or drinks or fights.
-Counselors: Instead of having the encyclopedia give you all the knowledge you need without an in game interaction, make talking with people inside the court give you insight into clans, families, and enemies, based obviously on your relation with the lord or kingdom.
-Wife and children: When your son/daughter is born you would find the nest in the keep, with your wife next to it or a maiden taking care of him/her. Your wife gets happy when you visit and gives you a dialog that would improve or rest relation with her (you could be a total **** to her or be nice to her, just to give some character to yourself and your relationship).
-Feasts: I'm going to guess this is a missing feature that is yet to be added, but they're definetly required. Also, make feasts and feast tournaments pop up a quest from here and then. To give an example, in the short book of George RR Martin
The Mistery Knight there's a tourney and an enemy of the throne tries to start a rebellion. Events like this can highly improve the experience.
-Allow you to ask the location of the lord to the guards: Instead of finding where a lord is telepathically, you would ask the guards just like in Viking Conquest, its really weird and tedious getting into the encylopedia, finding the person you're looking for and then seek him out, the older way was a lot more engaging and you wouldnt be taken out of control of your character and party to look a wikipedia. I really hate being taken out of control to do this.
Taverns:
-Populate taverns: Taverns are the place of rest not only of common people but of soldiers who are visiting a town. Filling them with already occupied mercenaries who serve other lords would give the impression that people is coming in and out, allowing you to meet interesting characters, like other mercenary companies, mercenary leaders, drunk soldiers, drunk thugs, and tourney competitors, like I've said above.
-Your own party: You can find your own companions in town, which is great, but I would also add part of your party inside the tavern, only that you would find the most common part of your party in it. Talking with them would allow you to improve relation with them, buying them drink, and spending the night with your party getting drunk at cost of renown.
-Fights: In Warband you would find drunken people, I dont know why this isnt a feature here, as I've said, you could find mercenary leaders, for example if one were to pick a fight with you, you could fight him back at cost of renown or relations with the employer lord, same happens if you find soldiers of another party who are making trouble.
Special Companions:
Having random companions allows any lord to have companions without getting into trouble, however, they're very uninteresting and forgettable. Here's what I would like to see in this game
-Companions with a visible backstory: Every faction has 2 o 3 special companions, they live in a specific town, have a specific home/place inside these towns or villages, a wife/husband, children, brothers, etc.
-They require you to complete a quest in order to hire them
-They would talk to you about your decisions, leave your party or require convincing when attacking their own homeland, and request leave or help when their land is in danger (depending on what backstory the character has)
-They have midterm quests, lets say one needed money to help his little brother and joined your party, once he made a good ammount you would need to take him to his land and see him fulfill his quest, allowing you to improve relation with them and therefore improving their stats, or making them give you a gift, like an special legendary sword.
-If they die you would have to take account for their death, for example, you would have to visit his home and pay respects to his wife if you want to, improving renown.
This is all I can write now, I'll write more when I'm inspired, if anyone wants to contribute we could make a special post dedicated to immersion suggestions. Also if anyone writed one of these before please tell me so I can credit you. Thanks all for answering and keeping the post alive, this is very important to me.