OP, you don't have to give your brother a piece of the milk glass. It's yours and you can do whatever you want with it. And you don't need to feel guilty about it. Your mother gave it to YOU.
That said, if it were me, I'd give him a piece of it. It's not valuable, and it apparently wasn't particularly important to your mother, since she was willing to sell it at a garage sale. So the only sentimental value it has is whatever you have attached to it. Just unattach it to one of the pieces and be uninvested in it. Give it to your brother and be done with it. Do not think a thing about it again. I believe this would ease whatever twinges of guilt you are feeling, and it'll also get your brother and your mother "off your back." Not that they're ON your back, but figuratively speaking.
In the big scheme of things, it's not worth the space this is taking up in your brain or your heart. Detach from one piece emotionally and let it go.
Have fun at rummage sales trying to find a piece to replace it.
The last line here, unfortunately, spoils the concept.
If you are going to give it up, and not think about it, don't think about it. Trying to replace it takes up that space in the brain, and the heart, you mention. If you really can't give up the glass without thinking every time you see the set, "There's something missing there. I must work to replace it," you have not unattached from it.
Plus, I get a vibe from this letter that this is not an isolated incident. If Brother has a habit of not wanting something until his sister has it, and then deciding his life is Totally Ruined!!!1! unless she gives it to him, there comes a point when the most compliant person will dig in their heels and go "No, this is mine, and you can't have it!"