I have no problem playing a robotic/androgynous and/or asexual character in this game. Part of the role-playing.
Yeah, I don't see the logic in the argument that goes "I'm a female player, and demand my female PC has breasts, otherwise the DM is being sexist". It's a fantasy game - you can play a range of characters that can be very similar to you, or very, very different - some of them anatomically so. Why is the ideas of having non-Human properties on a species sexist?
BECMI used to allow Treant PCs. By this logic, only male players be able to relate to a Treant PC because they lack breasts. Or if this only relates to Humanoids, what about playing a Humanoid droid in a Star Wars RPG? Can neither men nor women relate to playing one because they lack the requisite body parts, even though droids can have a gendered personality (e.g. C3PO comes across as "male")? Doubtful.
Also, I wasn't aware that a creature cannot be regarded as female if it doesn't have breasts (in fact, that argument seems extremely sexist to me!).
That all said, I don't see a problem with Dragonborn having breasts. At first I thought the idea was totally stupid, but I've been persuaded by the many good arguments upthread.
As usual, I think it's very much the DM's call, and it should be decided one way or the other. Unless the DM is doing it to bait female gamers (and is therefore an enormous jerk and you wouldn't want to be in his campaign anyway) I don't see what the issue is whichever way the issue is decided. You're putting yourself into a foreign mind, exercising empathy, each time you roleplay. I truly don't see how a bit of physical anatomy here or there matters.