Did the man fall short in his duty towards the girl when he did not register her in his name, but rather placed her in an orphanage to be registered under another name, which led to her being lost?

Your friend did well by picking up the child and taking care of her. However, he is not permitted, according to Sharia, to attribute her to himself. If his wife breastfeeds the child five full times, she becomes their daughter through fosterage. It is permissible to place the child in a trustworthy and uncorrupted care home; otherwise, he incurs sin. Your friend has the right to claim the foundling as long as he is trustworthy and capable of caring for her, and his position becomes stronger if the child is his daughter through fosterage.