What is the assigned identification number for someone incarcerated

Incarcerated individuals use their Social Security Number (SSN) or Individual Taxpayer Identification Number (ITIN) for tax purposes. 

The IRS does not generally issue a special tax identification number specifically for inmates

For more detailed information, you can refer to the following IRS resources:

• Individual Taxpayer Identification Number (ITIN)
• Prisoner Re-entry Program
• SSN Changes

If you need something else reach back out.

You do not mention why you are asking for an ID for a prisoner.   You will not be able to claim someone who is incarcerated as a dependent.   Their support is being provided by the corrections facility they are in; any money you are providing for incidental expenses, etc. is a gift, which is not deductible.