Great topic! I'm glad it was brought up to the top again.
I am neither a single parent or a birthmother who matched with a single parent but I am the child of a single parent. My mother remarried when I was nine but until then she did it all by herself.
While I am, obviously, the same gender as my mother she tried to do activities with me that she thought father's would do with their daughters also. She didn't want me to lose the "masculine" influence that a father undoubtedly has on his daughter. We played catch, went camping, hiking, fishing, rode bikes, caught frogs, wrestled, played sports, etc...along with all kinds of girly tea parties, dressing up, shopping trips, coloring, reading, watching Shirley Temple movies, etc... As a result I have a pretty cool mom who can dress like a princess OR pitch a tent in less than five minutes.
I also had a lot of positive male role models in my life. Coming from a very close knit family I was never short uncles who loved me and treated me the same as their children (especially my Uncle Al who had only sons, he loved to spoil his neice-daughter!) My grandfather also taught me to shoot guns, catch chickens, work on cars, and the names of tools.
I think my advice here would be that single parents include influences and role models in a child's life that give them the opportunity to experience both genders of parents. There is no reason why a single father couldn't play tea party or Barbies and sports with his daughter or why a single mother couldn't teach her son to catch a ball and read picture books together at bedtime.
Being a well-rounded single parent shows the child that stereotypical general roles are a thing of the past and that males and females are equally capable of most things. And having a strong family unit with male role models for your child exemplifies the proverbial, it takes a village to raise a child. And that sense of community can only help your child's self-esteem and well-being.