Yes, they can to devastating effect, however they prefer not to.
First the easy part: A succubus/incubus has the ability to charm a creature for a period of 24 hours, and the target under this effect does whatever the succubus/incubus says, up to and including suicide (although it does get a saving throw before performing a deliberately harmful action.)
Source: MM pg. 285
Charm. One humanoid the fiend can see within 30 feet of it
must succeed on a DC 15 Wisdom saving throw or be magically
charmed for 1 day. The charmed target obeys the fiend's verbal
or telepathic commands. If the target suffers any harm or
receives a suicidal command, it can repeat the saving throw,
ending the effect on a success. If the target successfully saves
against the effect, or if the effect on it ends, the target is
immune to this fiend's Charm for the next 24 hours.
The fiend can have only one target charmed at a time. If it
charms another, the effect on the previous target ends.
Second: They are of the creature type Fiend so they meet that requirement.
As for the preference I mentioned above, that actually falls under the flavour of how a succubus/incubus acts in some realms. There's a rather involved write up which details how the succubus/incubus goes about seducing a victim, and the focus on how the succubus/incubus wants the victim to succumb willingly. The goal of a succubus/incubus is to get the victim to give in to temptations that they otherwise wouldn't. Because of this succubi and incubi like to target virtuous individuals as their souls are the sweetest once corrupted.
The succubus or incubus resorts to charming a victim
magically only when necessary, usually as a form of
self-defense. A charmed creature isn't responsible
for its actions, so forcing it to behave against its will
won't bring the fiend closer to the ultimate prize: the
So as for your question: Yes, it's entirely possible for you to have charmed a target that then became wholly infatuated or dedicated to you. Nothing about the charm effect indicates you know it was magical or that you know the succubus/incubus did it to you, so if you are playing a character who is "possessed" by a succubus/incubus it could simply be a character who has been targetted and is routinely visited by said creature.
How would the succubus/incubus accomplish this task daily? Etherealness or shapechange, it possesses both abilities.
Etherealness. The fiend magically enters the Ethereal Plane
from the Material Plane, or vice versa.
Shapechanger. The fiend can use its action to polymorph
into a Small or Medium humanoid, or back into its true form.
Without wings, the fiend loses its flying speed. Other than its
size and speed, its statistics are the same in each form. Any
equipment it is wearing or carrying isn't transformed. It reverts
to its true form if it dies.
It can either visit its target when it's alone using etherealness to revert back to the material plane in order to charm it's target, or it could simply shapechange into a small or medium humanoid, like a damsel in distress or a small child looking for it's mother and fleeing monsters. Either way, it can actively charm it's target without giving itself away effortlessly.
Succubus and Incubus are one of my favourite monsters in the game because of how devious and devastating they can be when wielded right. I hope you enjoy playing your character. Having a succubus/incubi follow you about acting as a possessing element is very interesting and creative.