Do people who work in Korean enterprises count as seniority if they don't pay social security?

Count. As long as the salary is paid normally, it is equivalent to working in this enterprise and counting the length of service. You can ask the enterprise to give yourself social security, otherwise you can report the enterprise breaking the law.

Social insurance refers to the non-profit social security system with income redistribution function that the state participates in to prevent and force most members of society. Social insurance is a social and economic system that provides income or compensation for those who lose their ability to work, are temporarily unemployed or suffer losses due to health reasons. The social insurance plan is organized by the government, forcing a certain group to use part of its income as social insurance tax (fee) to form a social insurance fund. Under certain conditions, the insured can get fixed income or loss compensation from the fund. It is a redistribution system, and its goal is to ensure the reproduction of material and labor and social stability. The main items of social insurance include endowment insurance, medical insurance, unemployment insurance, industrial injury insurance and maternity insurance.