Wolverine, Logan, or James Howlett has a long history. Until recently (in comics) not all of it he could remember. He had accumulated different people that he viewed as family and at least two that are technically are. This is not a comprehensive list, just enough that you get an idea of the nature of his idea of family.
I am going to start with his biological son Daken. His mother was a Japanese woman named Itsu, whom I admit to not know that much about. The important part is that Daken grew up not knowing his father and learning to hate him. Thinking that he abandoned his family for one reason or another. In reality, he had thought both had died.
Daken drifts back and forth between being a villain and an occasional hero. He was on the team of Dark Avengers pretending to be Wolverine. Even taking on different missions with the hope of getting a chance to kill his father. I’m not 100% caught up, but last I understand before Wolvie’s temp death, they had a working relationship.
Like his father, he has claws, though his have two on top of the hand and one from the bottom wrist. His were later fused with a legendary sword Masamune which halts the healing factor of people with super healing. A sword that appeared to have killed a crazed Sabertooth. On top of his abilities he got from his father, he also has a form of pheromones that allows him to manipulate, not control, other people’s emotions.
There are a few notes worth mentioning about his use of that ability which I am not going to discuss here. Maybe another article.
His ‘daughter’ is the female clone of him known as X-23 or Laura Kinney. She was created when the Weapon Plus program (Weapon X is the tenth incarnation of that program) wanted to recreate their greatest success. They were only able to salvage the X chromosome and made a female clone.
She has all of the abilities of her ‘father’, but not the training, discipline or experience that he does. That is not to say she is a pushover by any stretch. She is still probably one of the most dangerous people. Worth noting about her claws is that they moved the middle claw on both hands down to her feet. So she can kick stab you.
Another thing that needs to be mentioned is a former weakness of hers. X-23 had a bit of her own brainwashing that was triggered by a specific smell. If she smelled it, she would immediately go into a berserker rage. Killing any who were near her. This is what killed her surrogate mother when she was unknowingly doused. This has been removed by Professor X or another psychic, though it could resurface.
Wolvie also has two ‘adoptive’ daughters. They viewed him as a mutant father figure and went to him for regular advice and protection. One is Jubilee that I can only describe as shooting sparkly explosions out of her hands.
When she first developed her powers she developed a bit of self-hate and didn’t want to be at the X mansion. Logan went out of his way to show her affection and making her feel at home. Eventually she became one of his daughters that he took care of. Giving her additional training. The most memorable moment is when she took his cigar and tried to imitate him. Only to throw up.
He justified smoking because of his healing factor. Told her he never wanted to see her doing it again.
His other daughter, one that was more delved into, was with Kitty Pryde. He taught her how to be a ninja. That is not an allusion to anything, he literally took her to meet and later kill one of his old masters. There she learned how to fight and move like a ninja. Making her ability to pass through walls even more dangerous.
There is a whole graphic novel that delves into his time in Japan with her. During that time he fights his old master Odgen and she becomes a strong fighter on her own. He continued to be the person she went to for advice and counsel. In the Marvel universe, almost everyone has been trained by Captain America or Wolverine at some point.
Speaking of, Black Widow. Now she does not view him as a surrogate father. She instead thinks of him as an uncle. During some of her early time in training and during his drifting days, (btw she is older then she appears), he spent a great deal of time making her into one of the most dangerous women. Even to this day calling him her uncle.
These are just the people we know about and the ones in the 616 universes. I could talk about the Old Man Logan universe as well (humorously number 90210), but I would prefer you read that one.
If you’ve been watching Marvel movies and want to see what a bad night could result in, that graphic novel does it.