Madame Alvarez's life as a courtesan hasn't resulted in much to live on. Translation: she's poor.
In Colette's novella, we're told that Madame took the name of a Spanish former lover. It's significant that she uses "Madame" i.e. "Mrs." Her apartment, where she lives with her daughter and Gigi, is shabby-ish but cozy. She putters about making tea and reminding Gigi to keep her feet off the table. In many ways, she's the most rational, sensible character in this story, the one who only wants the best for her little Gigi. She's gracious, well-mannered, and low-key, unlike her flamboyant sister.
Madame Alvarez's story is one of regrets: though she was in love with Honoré Lachaille when she was a young woman, she left him when he strayed. Her daughter is a struggling, self-centered opera singer who has no man to support her after she refused to be the mistress of a wealthy old guy who owned a bunch of flour mills. Gigi's mother never took to "the life." We never see her onscreen.
This grandmother's biggest hope is to see Gigi brought up in a proper way and taken care of by a man, preferably one who loves her, although she'll settle for rich. Mamita's a good friend of Gaston, which is a strange relationship given their social-class differences. (In the novella, it's suggested that she knows Gaston because she was involved with his father for a time. In the film, that relationship is transferred to Gaston's Uncle Honoré.)
Madame Alvarez is, like Aunt Alicia, totally old school, if by "old" you mean the "oldest profession." Both believe that a lady needs financial stability and a man to take good care of her. If it's the right amount of money, and the right kind of man, everything else will take care of itself.
But where do Madame Alvarez and her loopy, extravagant sister diverge? It's clear that the everyday mothering and care for Gigi has given Madame Alvarez a slightly different set of values. For example, after Gaston's gone gaga over Gigi at the beach and wants to take her to tea, Madame Alvarez protests:
MADAME ALVAREZ: If it were just you and I, Gaston I would say, "Take Gigi wherever you want." But there are others, Gaston. You are known everywhere. For a woman to go out with you alone now, with the eyes of Paris on you […] Let us say she would be labeled. A young girl who goes out with you is no longer an ordinary young girl. Not even a respectable young girl.
Gaston doesn't like thinking of himself as a guy who Gigi needs to be protected from. But Mamita holds out for a respectable offer for Gigi, i.e., a commitment from Gaston to keep her as his official mistress in a lavish apartment with all the clothes, horses, and servants she might need.
MADAME ALVAREZ: I understand responsibility to Gigi better than you. I'll do all I can to entrust her only to a man who's able to say, "I'll take care of her. I'll answer for her future."
Wait, what? That's being protective of Gigi? Making sure she's a kept woman? Well… yes. In Gigi's world, that's the best she could hope for, and Mamita knows it. As Gigi realizes, the women in her family don't marry. So when Gigi first rejects Gaston's business arrangement, and he storms out, Madame Alvarez yells after the crying girl, "Gigi have you gone out of your mind?"
But perhaps this has more to do with shock than anything. Because a moment later, when Aunt Alicia comes over to berate her sister, wondering why she hadn't stressed to the girl the delights of such an arrangement, Madame Alvarez confesses that she had none to offer. "Couldn't you have invented them?" Aunt Alicia asks. "I could not," Madame Alvarez. She couldn't sell the deal. She's not going to make stuff up just to get Gigi to cave. She knows Gigi's not interested in money and reputation.
AUNT ALICIA: Perhaps I should talk to her and tell her what she's missing. It's the glory of romance, forgetting everything in the arms of the man who adores you…listening to the music of love in an eternal spring.
MADAME ALVAREZ: And when the eternal spring is over?
AUNT ALICIA: What difference does that make?
MADAME ALVAREZ: It makes a great deal of difference to Gigi.
Ultimately, Mamita has to acknowledge what Gigi wants even if she thinks she's making a big mistake. Mamita gets her happy ending and so do we.