I had never heard of this but it is a fascinating topic:
To specifically answer your question was this specific to a specific tribe or clan, no it was more wide spread.
The book is referred to as a JokBo (族譜, 족보) and is managed and maintained by the eldest son and it is partially related to the practice of Jesa (ancestor worship up to 5 generations). Though it is also about a sense of knowing and maintaining and knowing ones culture, but also proof of 'class' to a degree (but less so these days) from my quick readings about it.
I believe the claim of 5000 years may be a mis-interpretation of the date structure within the book (partially due to lifetime of paper as well as knowledge of writing) and from the investigation I was able to do it seems to be that the oldest known copies of these books date back to the 12th & 13th Centuries A.D.. Also of note, the two separate Korea's is historical quite new and so it is a practice that was practices throughout.
The practice of maintaining them, especially once migrating outside of Korea though I pick up may be waining due to references to challenges of people returning or visiting to Korea and it can be embarrassing for them to know know their families history.
The site KoreanGenealogy.org seems to be an awesome resource for anyone investigating and they have an book online for understanding and reading JokBo's for English Speakers.
Examples: One Two Three
Note: I am not Korean and so I basing my answer based on reading some articles not on personal knowledge.