EDIT. Upon request of @schematic_boi, let me try to clarify. My original post only gives an example where the motive does not determine the variety, conditionally on a conjecture of Orlov. In fact, I realized that there are simpler and unconditional examples: if $S$ is any scheme, projective bundles of given rank over $S$ give rise to isomorphic motives (in fact just Tate motives) in the triangulated category $\mathrm{DM}(S,\mathbb{Z})$. This is the projective bundle formula, for this level of generality see Cisinski-Déglise, Triangulated categories of motives, 11.3.4. But I think this won't answer @schematic_boi's first question, because the topological fundamental group of a projective bundle should be isomorphic to that of the base. On the other hand, Schnell's article The fundamental group is not a derived invariant mentioned by @user25309, together with Orlov's conjecture, provides a conditional answer, at least with $\mathbb{Q}$-coefficients.
Original post. Orlov has conjectured [1] that two schemes having equivalent derived categories of quasi-coherent sheaves give rise to the same motive in the category $\mathrm{DM}$. In particular this would imply, due to a result of Lesieutre [2] that there are infinitely many non-isomorphic smooth projective 3-folds which give rise to the same motive.
References:
[1] Orlov, Derived categories of coherent sheaves, and motives. Russian Math. Surveys 60 (2005), no. 6, 1242–1244
[2] Lesieutre, Derived-equivalent rational threefolds. Int. Math. Res. Not. 2015, no. 15, 6011–6020.