To clarify what's happening, let us introduce the etale Grothendieck ring varieties $K^{et}(Var/k)$ by imposing the scissor congruence relation AND the relation $[X] = [F][Y]$ for every finite etale covering $X \to Y$ with (finite) fiber $F$.

Let us prove that at least when $k$ has characteristic zero, we have $K^{et}(Var/k)$ isomorphic to $\mathbb{Z}$ via the Euler characteristic with compact supports.

The proof goes in three steps. First we show by induction on dimension and using Noether Normalization that every element of $K^{et}(Var/k)$ is represented by a polynomial $P(\mathbf{L})$ in the Lefschetz class. The key observation is that any $n$-dimensional variety $X$ has an open subset $U$ which is a finite etale covering of an open subset of a projective space $U' \subset \mathbf P^n$.

Then we show that $\mathbf{L} = 1$; this follows using the $2:1$ covering $\mathbf G_m \to \mathbf G_m$, $z \mapsto z^2$ as in Jim Bryan's comment. Thus every element in $K^{et}(Var/k)$ is represented by an integer.

Finally, since the Euler characteristic respects etale coverings, it gives a well-defined homomorphism $K^{et}(Var/k) \to \mathbb Z$, and we just showed that this is an isomorphism!