Is work a vector quantity in physics?

[PeroK]

So the idea is that one is using R as the set of elements of the vector space and R as the set of scalars.
With ordinary real addition for the addition of two vectors and ordinary real multiplication for the product of a vector and a scalar.

Given that, I do not see that there is any freedom to define anything non-obvious. Everything is already nailed down.

Every n-dimensional real vector space is isomorphic to $\mathbb{R}^n$. You are right, there are no other options.

There is flexibility when you define an inner product or norm.

The starting point for physics, I would say, is that $\mathbb{R}^3$ is a real vector space equipped with the usual inner product, leading to the usual Euclidean 2-Norm: $|v| = \sqrt{v_1^2 + v_2^2 + v_3^2}$.

When we talk about the magnitude of a vector, there's not any doubt we are talking about the usual 2-Norm.

It's not entirely clear, I guess, when elementary kinematics is taught in one-dimension exactly what is assumed, as an underlying mathematical framework.