Every geisha wants to be her, and every man wants to be with her. Fate may often be cruel to Sayuri, but it was kind by matching her with Mameha as her older geisha sister. Sayuri is in good hands with her mentor, who teaches her how to bow, how to pour tea, and how to seductively reveal the inner part of her forearm (le gasp!) to titillate the men.
Mameha is basically the Tyra Banks of geisha. She trains Sayuri to be Kyoto's Next Top Geisha, and even teaches her to smize, telling her to practice what she says with her big, beautiful eyes.
But we wouldn't say Mameha is a friend. She's the closest thing a geisha will ever get to having a friend, but she remains firmly in the mentor camp. We find out late in the book that Mameha took on Chiyo as a little sister at the request of the Chairman. She also has no qualms about using Sayuri to teach her rival, Hatsumomo, a lesson. And she makes a pretty chunk of change by betting Mother that Sayuri will be a success.
So there is always an element of superior/subordinate to their relationship, instead of the equality of friendship. And after the war, there is a bit of a role reversal, with Sayuri usurping the throne as Top Geisha and requisitioning Mameha for jobs. But because Mameha is not prone to bitterness and jealousy like Hatsumomo, she respects Sayuri and helps her.
Because this book is Sayuri's memoir, we rarely get to see the characters independent of her. And geisha, being notoriously secretive, don't even talk to each other about their feelings. But they do have feelings, and Mameha is deeply affected by her danna, the Baron, although she doesn't show it.
Mameha doesn't have feelings for the Baron, she says that much. But she does feel sadness and regret for abortions he makes her have. At a local shrine, she builds three jizo statues to honor "the three children she'd aborted at the Baron's request" (28.67). Sayuri only mentions these small idols in passing, but it is clear they are important to Mameha, and the loss sticks with her. They were the only shot at a family she ever had.