Is Unemployment Insurance the same thing as Disability Insurance?
NO.
Unemployment insurance is an assessment every employer pays as a percentage of their payroll. The rate will vary with experience (i.e. an employer with high employee turnover might have a higher rate than an employer with very low turnover).
Workers Compensation is a policy every employer must have in order to cover their employees in case they are injured on the job.
Disability Insurance is private insurance that would replace (a portion of) one's income if they are too sick or injured to work. It can be an employer based group policy, or an individual policy that would cover an insured even if they switch jobs. NY State also has something called DBL, which provides a minimal disability benefit (IINM something around $170/week for up to 26 weeks) which is paid by employer, who may in turn also bill a small portion to the employee.