In vanilla, Increase relationship with them high enough, and then this option will appear. But it also requires 3 interest points, so you will need a bit for it to recover.
This will allow to recruit them in your clan, not your faction, BUT they also automatically follow your to faction (and if you are member of multiple due to "training" journey, then you can make a choice if they should join current one), thought if you purposefully put them into your home in some other city, they will mostly chill there, taking care of your clan affairs, if you order them.

Marriage or brotherhood oat is another way to get same results.

There is no automatic method to hire them for the faction, by asking other general to do it for you. Unless they are your descendants coming to age or some events. So for that cases you just need to hope that they will travel into one of your city and then you can just sent someone to recruit him or do it yourself.

(And raid the city to destroy all taverns so they leave for others is a jerk move).

Also, there are some politics that increase the chance of people just directly go to you and ask to join. They require you to already have a character with such politics in your faction, and then enable it in the screen which you'll get if you press strategy points as a leader.

By the way, you can't place officers in all of your party slots without mods, i think it's only 2 or 3 slots in vanilla? And they are shared with mercenaries (but beasts are separate). And all of them, officers, mercenaries and beast, use the same deployment limit during battles.
Althought there are some stuff that changes if you have extra officers with you, like during team tournaments it's them who will be in your team, instead of you joining some random one. Or when you ask for duels or tactical battle for entertainment, you will get one of them participate on your side, and your opponent will call some of his friends, which allows you to impress that friend too if you did a good job there, and not only your original target.

And when officers are of your clan or your friends, or family, you can ask them to follow you for a bit. Depending on what type of relatino you have, the lenght they follow you will change from 1 month to until you manually let them go/redeploy into one of your homes.