@Padlom While the man does know her name, he's not sure about her identity (page 2). He's just meeting a girl here and prefers to confirm if she's indeed that girl. Moreover it seems that it's not his first time bringing a girl here as he seems to know the area (next pages). There is also the title of this work: Enkou / Enkō = Enjo kōsai = compensated dating. Hence I would say that the girl and the man have no relationship with each other.
She calls men "uncle" and men know her name. But 小父さん or おじさん familiar language Often written using only kana. Referring to the old man, Mr. If so, she might be working in prostitution.