What does Williams mean in "In Memory of an Aristocrat" when he writes: "There is only one true aristocracy. . . and that is the aristocracy of passionate souls"? Explain your answer using examples from "In Memory of an Aristocrat," "The Glass Menagerie," or both.

In reference to your quote from "In Memory of an Aristocrat," it is very likely that Tennessee Williams was referring to bohemian artists like Irene. The word "aristocrat" is associated with distinction and refinement. Williams's hypothesis is that the working-class artist has more to offer the world than the unimaginative aristocrat. Thus, the "aristocracy of passionate souls" consists of all who purposefully challenge the bounds of social decency through their promotion and creation of experimental and provocative art. These "passionate souls" challenge the status quo and so, are the true "aristocrats" and leaders of society.
In his time, Tennessee Williams was much influenced by Chekhov, who himself highlighted the virtues of the common man. Similarly, in many of Tennessee Williams' stories, the aristocrat classes are often portrayed in a negative light, while the working classes are often shown to be imaginative, open-minded, and socially aware in contrast. This is true for both "In Memory of an Aristocrat" as well as The Glass Menagerie.
In "In Memory of an Aristocrat," Irene is the bohemian artist and prostitute who lives to challenge the conventions of classical art. She navigates life on her own terms, and is as free and adventurous in her sexuality as she is in her art.
For the Annual Spring Display, Irene chooses to submit ten of her best works. The text tells us that her art is poles apart from those normally accepted by the Annual Spring Display in New Orleans. The event, after all, is sponsored by select artists who move only in the most prestigious of circles and live in studios "sparsely furnished with very beautiful things, great oval gilt-framed mirrors and inch-thick Oriental carpets."
In the end, all of Irene's art pieces are rejected; they are considered too provocative for public consumption. Furious, Irene makes a scene at the convention by displaying the largest of her art pieces herself. The narrator tells us the image of Irene storming the event (kicking and screaming obscenities) is antithetical to the image of gentility on display. It is quite obvious that Tennessee Williams admires Irene's raw energy and emotional honesty, though. She is what he terms an "aristocrat of the spirit," a passionate soul who celebrates human expression in all its forms.
That same admiration for the passionate soul is evidenced in The Glass Menagerie. In the last scenes of the play, Tom decides he will no longer allow his mother's aristocratic pretensions to cloud his future. While his decision to leave his dependent mother and sister is controversial, Tom's character exhibits a universal human desire for freedom.
Tom means to investigate life on his own terms; he means to judge his actions, not through the prism of conventional expectations, but through the lens of his own perceptions. In leaving, Tom demonstrates his desire to savor the opportunities life throws at him, no matter how uncertain and treacherous they initially appear. He wants to live in an exciting world "lit by lightning" rather than in a sterile world circumscribed by the fragile and illusory light of candles. In this way, Tom is similar to Irene: both are members of the "aristocracy of passionate souls," determined to live life on his own terms rather than those determined by others.