The simile "like stone" reinforces the imperious message and tone of the poem. The speaker has made it abundantly clear that she's just fine being alone. She's one of those people who prefers her own company and thus has no need to admit anyone else into her little world. Even if an Emperor turned up one day and knelt upon her mat, the solitary speaker still wouldn't let him in.
That said, the speaker will let the occasional person into her life one at a time. When she does this, she promptly closes the door—or the "valves of her attention" as she calls them—on the rest of the world. There's room for only one person at a time in the speaker's life, and that's how she likes it.
Returning to the stone simile, we can now see how appropriate it is. Stones are cold, hard objects, and we get the distinct impression that the speaker, in shutting practically the whole of humanity out of her life, is indeed cold and hard. The abrupt way that she deprives the rest of humanity of her attention after choosing one, and only one, companion, comes across as almost inhuman. Once again, the image of the stone—cold, hard, and firm—springs to mind.