Scroll down to the second code block if you don't want to read my rambling.
Models in multiplayer are technically the exact same in singleplayer, however since singleplayer has much more going on with playermodels, such as cutscenes and customization, the ways you get them to work properly are a bit different.
Typically skins in JKA are only made for multiplayer, because the playerbase was always larger for MP compared to SP. Many modders just didn't bother giving SP support to their skins/models due to it adding bloat to their PK3 file, or perhaps just not knowing how to. For years other modders had been making separate mods that allow you to play as certain skins in SP, meaning you had to have two PK3 files for one mod, which is a bit silly, however I made so many of those mods in my early days, it was a good learning experience.
Instead of a mod, you can just use a command in the console to change your skin. You've probably already tried to use the playermodel command, which actually only changes your appearance to an NPC, which means you also get all of that NPC's stats and weapons. So if you want to play as Luke, you'd do playermodel luke and also get his 200 force points, all of this force powers, his saber, and sounds. That probably sounds cool, but the two biggest downsides here is that you won't be Luke in any of the cutscenes, nor will you be Luke in any future levels until you put that command in again.
So how do you do it properly? You'll use the same command, but with a few additional parts to it. The playermodel command has the ability to customize your character in cases that it's properly set up for multiple parts, for example the default Jaden models have multiple heads, torsos, and legs, which can be set via his command. But it also works with literally any player model too. Here's an example of customizing a Jaden model that is already in the game:
playermodel jedi_hm head_a1 torso_a1 lower_c1
This would set your player as the blonde Jaden with the Jedi tunic and tan pants. But that can be done in the menu, so that's not really that big of a deal, unless you were wanting to change it mid-game and didn't have my ingame character menu mod.
For models that don't have SP customizations, you can simply do this:
playermodel luke model_default model_default model_default
Replace luke with the model you are wanting. If it's a mod, the author of the mod should have put the command for it to work in MP in their readme or file description, for example model kylo_ren. So just go by that when using the above command. You'll always be able to use model_default no matter what. You could also do model_blue or model_red if the skin has team color support.
The more you know about PK3 files, the easier it will be to know what variants you can use with this command. You can look in the model's folder to see what .skin files there are outside of the three main ones. For example, a kylo_ren folder may have model_nomask or model_hood or other things like that. If you have no idea how to open PK3 files, you can get started here, and if you're interested in getting started with modding in general, here.
You can also bind this command to a key, so you can change your appearance at any point in the game. I remember I used to change the appearance with the Jedi Customization mod, depending on the mission. If it was a infiltration-style mission, I'd set it to have full robe and hood, or a boss fight then have no robes, just like the movies.
Keep in mind that adding SP menu support for your characters is still recommended if you're a modder, and if you want to learn how, check out this tutorial.