I am only familiar with the Korean arts such as TKD and TSD. They number their belts as to seniority such as the Dan Bon system in Soo Bahk Do and the Kukkiwon. Is this a Korean thing or is it the same in Karate? Like if I get a Shodan from an organization or a school, is there a number associated with it?
There's not much uniformity in Karate styles, but black belts are somewhat the same.
Kuru Obi (Black Belt) is 1-10, with 1 being the lowest level of black belt. All black belts are Yudansha.
1st Dan - Shodan.
2nd Dan - Nidan.
3rd Dan - Sandan (I am a sandan, have held the rank for five years now). Sandan is often the first black belt rank considered to confer the title of 'Sensei' or teacher.
4th Dan - Yondan. Note that in Japanese, 4 can be 'shi' or 'yon'. We avoid the use of 'shi' when talking about people, because 'shi' also means death. Thus, no shidans.
5th Dan - Godan. Sometimes but not always, 5th dan and up begin to also have honorifics attached, such as 'renshi', 'kyoshi', and 'hanshi' - there are others as well. Not all organizations use them, they don't always mesh up with a given dan rank, and it's usually not automatic - a person can be a Godan and NOT a renshi, or they can be both. It depends.
6th Dan - Rokodan. In some systems, a Rododan is entitled to wear a different type of belt - such as a red-and-white panel belt.
7th Dan - Sichidan.
8th Dan - Hachidan. In some systems, a Hachidan is entitled to wear a solid red belt.
9th Dan - Kudan.
10th Dan - Judan. Often but not always head of system or organization.
Typically but not always, there will be only one 10th Dan in an organization. In modern times, this tends to be less true. There are a lot of Judans these days.
Other than belts being numbered 1-10, not everything is the same in every system, organization, or even between dojos. My training is in Isshinryu, and really can only be considered correct in my own organization, the WUIKA. Other Isshinryu organizations do things differently.
Typically, promotions are the number of your belt plus a year at that rank. So a Shodan would be considered eligible for Nidan in two years as a Shodan, or 1+1. A nidan would be eligible for sandan 3 years, or 2+1, and so on.
Black belts who are sensei can usually promote students up to one belt under their own rank, except for 10th dan. In the past, this has left organizations without a 10th dan, if theirs passes away or retires. Sometimes a system or organization has to step in to make a new 10th dan promotion to keep one at the head of the system. Some argue the legitimacy of promotions by committee, but it is what it is.