An answer has already been given in the [comments](https://mathoverflow.net/posts/comments/889725): the definition of $K_0(V_k)$ includes the relations $[X]=[X_\text{red}]$ so fat points are equal to points by definition. The intuition is that $K_0(V_k)$ was primarily conceived to study phenomena involving classical topological invariants (Euler characteristic of the underlying topological space, Hodge structure, etc), or point counts over finite fields, which are all insensitive to the scheme structure anyway.

There is however a theory which encodes richer data, though it (at least a priori) depends on more input than just the scheme structure of $X$, and is only defined for certain kinds of schemes. This is the notion of a motivic vanishing cycle attached to a pair $(M,f)$ where $f\colon M\to k$ is a regular function on a smooth variety $M$. The intuition is that this should be thought of as a motivic invariant attached to the degeneracy locus $X=\{df=0\}$, thought of as a subscheme of $M$, which is sensitive to the scheme structure of $X$. In your example, we would consider the pair $(M,f)=({\mathbb A}^1, x^3)$ which indeed has the double point $X$ as degeneracy locus. This motivic vanishing cycle takes values in a ring which is larger than $K_0(V_k)$, remembering also monodromy data about the function; in the example at hand, you would need to worry about order-three monodromy. But there is still an Euler characteristic map to the integers, which returns the answer $2$ for this example, indicating that indeed we are talking about a double point. 

The motivic vanishing cycle was defined by Denef–Loeser, and the interpretation I gave above arose in works of Joyce and Kontsevich–Soibelman in connection with motivic Donaldson–Thomas theory. A starting reference might be On motivic vanishing cycles of critical loci, [arXiv:1305.6428](https://arxiv.org/abs/1305.6428), by Bussi–Joyce–Meinhardt.