I do agree with you, it is absolutely embarrassing. And it's horrible that the state requires it.

However, (welfare requirements aside) from the perspective of the child, their right to know who their father is supersedes any other person's rights. A close friend of mine is in her 40s and she has no idea who her father is, and her mom won't tell her anything. They are no longer talking because she is in the process of doing a 23 and me to get some information. Please encourage your friend to find out who the dad is. Not because TANF requires it, but because her children deserve to know their history. And they also deserve the financial support of the sperm donor.