I'm a duke of the kingdom Lydia, and I'm physically located in some other king's land. Can I take over his kingdom but in the name of my kingdom? Would it have to be a kingdom union? Future thanks!
You have several options to "take over" (if we stretch this term a bit) another kingdom
Your kingdom can open up one or more treasuries in the area of an enemy kingdom and either destroy their treasuries to disable their "influence" sphere or have more influence than them yourself (by getting more population in the village where your treasury is located and by having more active treasures inside it. Keep in mind that they like will defend themselves and retaliate if possible. Also you need catapults IF you want to destroy someone elses infrastructure. Also also you can only have one treasury per king and duke (so max 3 as a normal kingdom) unless you gather 10000 *active* treasures, what IS a lot for smaller kingdoms (or you union, see below)
Your king could offer the other king a duke spot as a kingdom got 2 (or potentially even yours if its more beneficial to him/the kingdom). He would have to reactivate his treasury again but after that 12 hours his area would belong to you.
You could do a mix of the two suggestions above, asking the other king to step down and become a governor so you can activate your treasuries there.
Last option I know of is indeed to union up. Im not 100% certain of the conditions to do so as its fairly new feature and i didnt play a king since its implemented but the rouch is as follows. Your kingdom and the other kingdom have to have a common border for at least 30 days (and iirc the spheres have to include the kings sphere aswell. so you as a duke couldnt open up a treasury on the other side of the map, wait 30 days and qualify for union) Then both kings will get the option to invite the other to merge both kingdoms into the bigger one (victory point wise). The king of the bigger kingdom will stay king, the smaller ones will become vice king (basicly both have the same rights, so no big deal imho) and they will be able to have 4 dukes for a total of 6 "royal roles" So it would be possible to do it this way, BUT it takes 30 days of sharing a border and it is very hard (intended to be impossible) to "un-union" so if somethign goes awry king and viceking are theoretically stuck in this kingdom (while everyone else could leave, but thats not always a pleasant option either.