Finite degree polynomials are vectors yes. Polynomials is a typical example you study when learning about linear algebra. Doesn't say anything about real functions in general though, I don't think any linear algebra course should make the analogies made in this article, that'd be confusing

I think they should.

You don't need to go into intricacies like metrics and approximations because that's more like analysis.

You can however talk about infinite dimensional vectors spaces and talk about projections onto finite subspaces.