Yes, your teen should file their own return. They can indicate that they 'are claimed as a dependent' on your return in the My Info section in TurboTax.
Unemployment is considered taxable income (although it is unearned income). They may be eligible for a refund.
Click this link for info on How to Enter a 1099-G for Unemployment or Paid Family Leave.
This article gives detailed info on Unemployment Income.
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