It is the first intransitive verb sense in the Merriam-Webster page you link, "to change direction or course".
We, at least here in Britain, often use veer when giving directions, meaning to change course slightly, to turn just a little to the left or right. I think you'll hear it from SatNavs as well. In this case it is metaphorical, of course.
In cases such as this, the metaphor concerns direction, but doesn't necessarily concern it changing. Essentially, this could be rewritten as follows:
Whether your furniture tastes tend more towards AllModern or Mr. Starck, there's a home styling situation for everyone in this game.