Long way back in history
There existed a 4th major clan, Kuzunoha (or Abe) clan. They had isolated themselves for centuries after the Heian era because of the heating political environment and the death of Kitsune Heir. The estate is in Shinoda forest in the Izumi Province (in Osaka)
Now, the relation of the Kuzunoha clan ancestors/ and present with others:-
Gojo clan :- Gojo clan ancestor, Sugawara no Michizane (one of three vengeful spirits) and kuzunoha clan ancestor, Abe no Yasuna (The Husband of legendary Kuzunoha, the famous Kitsune (fox spirit), and father of Abe no Seimei (founder of Kuzunoha clan)), both were part of high ranking scholarly class, people call them sibling in bond, hence, to respect that relation, Gojo and Kuzunoha clans are known as "sibling clans".
Kamo clan:- Historically, During Heian Era(the golden age of sorcery),the Kamo family - specifically Kamo no Tadayuki and his son Kamo no Mitsuyoshi were the official teachers of Abe no Seimei. However, Seimei (son of Abe no Yasuna and the fox-spirit Kuzunoha) became so talented that he eventually surpassed his teachers.
A Kuzunoha clan is bitter rival to the Kamo clan, representing "natural genius" versus the Kamo's "strict tradition." Though kuzunoha respects them as their teachers.
In real Japanese history, the Kamo clan were the undisputed masters of Onmyōdō (the way of Yin and Yang), and their two main "jobs" for the Emperor were:
1- Blood and Exorcism (Which Gege Akutami turned into Blood Manipulation).
2- Calendar-making and Astronomy (The Rekidō).
Zenin Clan:- In real Japanese history, the warrior class (which the Zenin represent) and the Onmyoji (spiritual advisors like the Abe/Kuzunoha line) often clashed for the Emperor's favor. Zenins represent the rise of the Samurai dealing with strength, weapons, and seizing power by force. Meanwhile kuzunoha represents the Heian-era aristocracy including rituals, poetry, and influencing the world through unseen spirits. They were/are Political Enemies. The Zenin try to suppress the Kuzunoha's influence in the modern era, every time, calling their methods "outdated superstitions," while the Kuzunoha would look down on the Zenin as "thugs with Cursed Energy."
The Zenin clan mirrors the Taira or Minamoto clans (the warriors), while The Kuzunoha clan mirrors the Fujiwara or Abe (the court nobles).
Regarding the confusion between Abe no Yasuna and Abe no Seimei :-
Abe no Yasuna (The Father)-
He was the human who saved a white fox from hunters. That fox turned into a woman (Kuzunoha), they fell in love, and had a child (named Abe no Seimei). He was the one who formed the "Sibling Bond" with the Gojo ancestor (Sugawara no Michizane). He represents the clan's connection to the spirit world through love and blood.
Abe no Seimei (The Son) -
Son of The Fox and The human (Kuzunoha and Abe no Yasuna). He is the one who officially founded the Kuzunoha Clan (He named the clan after his mother). He was the "Natural Genius" who went to school with the Kamo clan and eventually surpassed them, causing that 1,000-year-old rivalry. Because his mother was a fox, he was born with supernatural intelligence and powers. He became the most famous sorcerer in Japanese history. Basically, Satoru Gojo of Heian Era. He's the one who turned the family into a Major Clan.