Is the white fan with a geiko or maiko’s name on it similar to a calling card? When do they hand these out and who gets them? When I was in Kyoto in 2023 I went to a dessert cafe and saw a few of these on their wall.

The white fans are called Uchiwa (団扇), and they're gifts given to establishments and customers of geimaiko during the summer months as a form of advertising ^^