I'm not quite of your questions, but here are some thoughts:
"W-2 unemployment witholdings"
Unemployment is not withheld from the employee's pay, and is therefore not on the W-2. It is a cost paid by the employer.
"What are IDs"
When the employer pays for Federal Unemployment (Form 940), they use the Federal EIN of the employer. When the employer pays State Unemployment, the employer often uses a separate ID number, perhaps their State EIN, or perhaps an ID number specific to Unemployment.
"Isn't employer responsible for paying/holding and paying valid claims"
No. The employer pays the Federal and State the unemployment fee/tax, and the State makes the actual unemployment payments to the former employee. Although the State usually asks for information about the situation from the employer, the State makes the actual determination if the person qualifies to collect unemployment, and makes the payment to the former employee.
