General theory of third party servers then.
Games can do all sorts of things to effect multiplayer though there will be two main models
1) Multicast and all done on hardware.
2) The server actually does something.
1) Normally has the remote servers basically acting as matchmaking and maybe high score tables. After that the server might take data each person sends it and then sends it on to all parties (if you need to upload 100K a second of data then for an 8 person game that becomes 800 and thus breaks a lot of consumer ADSL, sending 100 to a big server which then uses its oodles of bandwidth to send said 800K/second is far more doable.
Alternatively you will send data to all parties or together make up a server after the developer run server does matchmaking, this is what is known as p2p servers as you send things person to person like you might with a torrent or something.
2) Has the server doing calculations on everything that is happening in the game, for instance you don't need to have an idea of every person on a given server for a mmo on your little device or in the case of some FPS where the server knows where all of the 64 players plus their vehicles, bullets.... and predicting effects where someone's connection will inevitably conk out for a second or two. World of warcraft is this but it was noted its requests were essentially a common database language and thus it was able to be pulled apart (plus possibly some leaked code or analysis of the game client -- between function names, observing what is happening on screen, being able to control what is happening on screen... you can learn a lot).
If there is a PC version where people can host their own servers, or at least poke and prod their connection more easily, then this is also an option for something to look at for some clues.
At the same time the stuff guily6669 notes is more how does it handle it.
For something like the DS it was noted that any secure data (more or less just a handshake) would also work non secured so that was intercepted, understood and the game tweaked (the s part of https was removed from the relevant urls) so when you changed the DNS (the thing that takes internet addresses and translates them to the current IP address for that site/address) and it pointed at custom servers it was all good (no worries about the security part of SSL being tripped when we had no certificates either). I am not entirely up on the wii stuff but I believe it is a similar story.
If however the game buried server names deep, or even IP addresses, vets security of data all the time you have to extricate all this which is a pain. Doable enough I guess but where the DS can usually have been done in seconds then this will take some serious time to do one game.
Local setups a la xlink kai.
Some games allow local multiplayer, common in handhelds for obvious reasons but with a lot of older xbox games being essentially PC games and LAN parties being a thing back then you saw LAN games also became a thing for the original xbox (and then some on the 360 as well but the concept was fading fast, not to mention a quirk I will cover shortly).
If you can take this multiplayer data and send it over a network then you can use a personal VPN (virtual private network) which essentially makes a local network over the internet and thus is obvious. xlink kai was then the name of a popular service for this for the original xbox, and a few other systems (still a VPN in the end though). The 360 frustrated this by including an upper ping limit that no normal LAN would ever likely see but all but a close internet connection on the same ISP in the same city would see. Said ping limit was able to be knocked out at the system level for JTAG/RGH systems (not DVD modded systems though) but nothing stopping people from burying it deep and making it hard to remove.
Said "if you can take the data" also becomes a thing. For the DS then the reason it never happened for local games was that the protocol it uses is quite weird and custom (a kind of headless wifi) which no consumer card really stands a chance of capturing. I imagine it is technically feasible with some of the really swish FPGA programmable radios some cards have or having some fun with a spectrum analyser (if not out and out custom hardware) but... yeah.
If said games support or otherwise generate a conventional wifi style access point then you can sit something in the middle of it (connect to your PC which obviously you can control the nth degree) and fire it wherever you like.
You also run into the problem that not all games feature local play or that local play is substantially different to online offerings and thus you may still want a proper replication of internet services, or maybe a more in depth hacking job for the games to open them up a bit.
Screen sharing internet setups a la kaillera
For those devices allowing splitscreen multiplayer then you can essentially stream the video and audio outputs somewhere else and take back control inputs. Sony and MS seem to offer something functionally equivalent to this with their various game session sharing services. kaillera is the name of one of the earlier methods for doing this, one which was supported in a variety of emulators.
If it is a device that is essentially separate and expects actual communications from something... outside its shell/controller inputs then this is not so useful as is. Some more recent efforts with handheld link cable games can still make use of something like this by emulating it all on one system, and people had been doing it for years with GBA emulators -- trying to send data that normally goes over a serial cable (very short, no disconnects/slowdowns and dropped packets normally there) and hoping the internet (the opposite of the last bracketed section) holds up is not a good plan. Obviously though this means emulating two (or more) devices at once on the same machine (or possibly LAN) with all the resources that requires. While remote control programs are good then for some twitchy/high speed reaction games it is rather more troubling, no reason not to try though and if you can sort a turn based game then no worries.
If you are doing it properly as well you might also want to hide screens for those games that need it -- I used to use the two emulators thing years ago to play advance wars on my PC, if we were playing with fog of war then it is not much good if I can just look across and see the second emulator and thus the troop types and positions (I solved it in my case by having a second monitor and controlling the other emulator with a controller instead of a keyboard).
You might also want a way to catapult saves and savestates around just in case things get disconnected.
For both the a la sub sections I should also link the following for while it is very old it certainly still applies
https://www.gamasutra.com/view/feature/131781/the_internet_sucks_or_what_i_.php
Legality of all this... tricky.
https://www.engadget.com/2018/02/20/esa-dmca-online-gaming-petition/
That covers an exemption to the DMCA (the main thing governing a lot of US software IP law, a horrible piece of law written by incompetents and people with a vested interest in making it suck if you ask me but a piece of law never the less) which allows third party efforts for abandoned games and as you can see there then the ESA opposes it even for abandoned things for reasons of competition, never mind current stuff.
At the same time then legal case wise then I have nothing really. There were plenty of private servers for WoW for years, before the wii online died then emulators could even connect to it, as far as I am aware the DS and Wii stuff has been just fine for years, the early* DS and Wii hacking work happened in public.
I am not sure what principles of law I would point to so as to make at least a moral case for all this -- probably some combo of the very same thing that allows you to ban me from your private network also allows me to set up my own. Anybody that joins mine already hacked and set it up to join and thus it won't bother your customers and their actions are a tacit agreement here anyway (anybody that can join a third party server here already knows what goes). Depending upon your methods of reverse engineering then it might tip over into the not so clean room methods but it is not like we care about that around here.
*technically it started at or around the same time that gamespy (which ran the service for the DS and Wii) was announced that it was going to be shuttered, obviously taking most DS and Wii games with it so eh.
I saw mention of cheats and hacks. Potentially a problem for some (if you can get to the point where you have third party servers you can have cheats/mods too) but no reason not to figure it all out and give people the option to implement things.
As far as working if you are banned already it should have no effect if done properly, I imagine there are half measures where people still use some official something but that is a half measure and would probably not be employed here. If you are banned already then obviously you are not going to be as able to help by using said banned console to monitor what sorts of data is sent.
Similarly if the bans are coming for essentially unknown reasons and you need a hacked device to figure out what is happening then you have a time limit there. If it is using local stuff then it should not matter at all -- the case where your device is blacklisted or has a "I'm banned" flag sent I don't imagine will be in play here.