I'll give it a shot.

We can use vector spaces for thinking about things that aren't primarily concerned with physical space like we are in Blender. Let's imagine something practical, if a bit absurd. Pretend we have unlimited access to three kinds of dough. Each has flour, water, and yeast in different ratios. What we don't have is access to the individual ingredients.

Suppose we want a fourth kind of dough which is a different ratio of the ingredients from the doughs we have. If the ratios of the ingredients of the three doughs we already have are unique, then we are in luck! We can make that dough we want by combining some amount of the three we have. In fact, we can make any kind of dough that is a combination of those three ingredients. In linear algebra, this is called linear independence.

Each dough is a vector, and each ingredient is a component. We have three equations (doughs) in three variables (ingredients).

This is a three dimensional vector space, which is easy to visualize. But there is no limit to how many dimensions you can have, or what they can represent. Some economic models use vectors with thousands of dimensions representing inputs and outputs of resources. Hopefully my explanation helps us see how vectors can sometimes be more difficult to imagine as directions and magnitudes.

It isn't structured using magnitude and direction, but as an element in a vector space.

It is just to consider polynomials and functions as vectors, and apply our meager intuition on 3d spaces. By introducing norms (size), you recover the "size and direction" analogy.

Check CompassRed's comment above.

The definition part of the wikipedia article has a table with these "nice relationships for addition and scaling". You will see that they also hold for many kinds of functions, such as polynomials and other more abstract things than points and directions in 2D or 3D. N-dimensional vectors for example, or using complex numbers, or both.

https://en.wikipedia.org/wiki/Vector_space