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Here is a sentence from a game that is based on interior designing:

Whether your furniture tastes veer more AllModern or Mr. Starck,
there’s a home styling situation for everyone in this game.

I know the lexical meanings of the word "veer", but am not sure if any of them fits here.

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.

veer is a verb associated with direction of travel. Either literally or figuratively.

• The car veered off the road.

That means: It suddenly went in a different direction that took it off the road.

• The conversation veered into a shouting match.

That means the conversation suddenly became a shouting match.

• His ideas veered into a dark place.

That means his ideas suddenly went to a dark place. (in the mind)

So, generally, it would be hard to imagine that someone's tastes (which are usually stable) veer anywhere....

• 
  Has your taste for vodka veered to whiskey? [suddenly changed] I guess you could say that....

The semantic trait for veer is to change direction suddenly or quickly.