It's true that a vector space has a designated origin. Whether you choose that to interpret as "vectors being anchored" is, I guess, up to you - that's just an issue of language usage, not of mathematics.

I personally don't consider functions or polynomials to be "anchored", but yes, of course there is the zero polynomial etc.

Also, keep in mind that physicists, for example, often use a more restricted definition of "vector" than mathematicians. That Wikipedia definition you quoted doesn't strike me as very mathematical. A mathematician's definition of an affine space is much more abstract.