EDIT: answer is expanded on OP's request.
The category $\pi Set_{fin}$ of finite, continuous $\pi$-sets is just the functor category $TopCat(\mathbf{B}\pi,finSet)$ where $\mathbf{B}$ gives the one-object topological groupoid associated with a topological group, and $finSet$ is the category of finite sets, viewed as a topologically discrete category. The groupoid $\mathbf{B}\pi$ is canonically pointed, so induces the functor $\pi Set_{fin} \to finSet$ i.e. an object of $Cat/finSet$. If you replace $TopGrp$ with the equivalent category $TopGpd_{1obj}$ of one-object topological groupoids, then this naturally comes with a notion of 2-arrow, and it is, as John Baez points out, conjugation by an element of the codomain. Then 2-arrows in $\pi Set_{fin}$ are natural transformations of functors to $finSet$.
Now really you are working with $TopGrp^{op}$, so you have two choices as to the direction of the 2-arrows, if you are taking 2-arrows as specified above. This should fall out of the definitions. Also, since $TopGpd_{1obj}$ is a (2,1)-category, you need to restrict attention to the (2,1)-category $Cat/finSet_{(2,1)}$ underlying $Cat/finSet$. Here $Cat/finSet_{(2,1)}$ is the isocomma category: objects are categories over $finSet$, arrows are triangles commuting up to a natural isomorphism and 2-arrows are natural isomorphisms in $Cat$ that are compatible with the 2-arrows in the triangles.
In more detail: given two objects $F\colon C\to finSet$ and $G\colon D\to finSet$, an arrow $F\to G$ is a pair consisting of a functor $f\colon C\to D$ and a natural isomorphism $c\colon G\circ f \Rightarrow F$. Given two arrows, $(f,c),(g,d)\colon F\to G$, a 2-arrow between them is a natural isomorphism $a\colon f\Rightarrow g$ such that the obvious 3-dimensional diagram commutes. Since all 2-arrows are invertible, from this commuting 3-d diagram we can write down an invertible endo-arrow $c+(G\circ a)+d^{-1}$ of $F$ (here $+$ is vertical composition of natural transformations--from right to left--$G$ really denotes the identity transformation on the functor $G$ and $\circ$ is horizontal composition).
I'm fairly confident that this is a 2-arrow in $TopGpd_{1obj}$. If one doesn't restrict to the (2,1)-category $Cat/Set_{(2,1)}$ then this doesn't work. (So really you should be thinking about $Gpd$-enriched categories, not so much general 2-categories.)
EDIT: Actually, it is easy to see that this is a 2-arrow in $TopGpd_{1obj}$, because the homomorphism induced by $(f,c)$ is conjugation by $c$ (with whiskering): $\alpha \mapsto c+(f\circ \alpha)+c^{-1}$.
Thus we have a 2-functor $Cat/finSet_{(2,1)}\to TopGpd_{1obj}^{op}$.
Now in going the other way, I think we actually need to use not just $TopGpd_{1obj}$, but the isococomma category $\ast/TopGpd_{1obj}$, where a functor between topological groupoids respects the basepoint up to a 2-arrow (which is an automorphism of the canonical basepoint). I haven't checked but this looks like the description of the functor in the preceeding paragraphs works better. Then the functor $\ast/TopGpd_{1obj}^{op} \to Cat/finSet_{(2,1)}$ is just exponentiation with $finSet$, and we don't have to think too hard about what the 2-arrows etc do.
Then you need to worry about the adjunction. But if you already have a 1-adjunction, half the work is done.
Two clarifications:
For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group.
While it's true that the motivic Galois group is a higher-dimensional analogue of the Galois group, it also should be true that motives are "just" a special kind of Galois representation, i.e. under the Tate conjecture the $\ell$-adic realization functor should give a faithful functor from motives to $\ell$-adic Galois representations, so the category of motives is the category of Galois representations with some restrictions placed on the objects and morphisms.
Of course these restrictions are highly nontrivial. Only for the irreducible Galois representations do we have a good conjectural description of which ones come from motives, via the Fontaine-Mazur conjecture.
So we can see that all 3 of these relate to Galois actions - the first two to Galois actions on fundamental groups, and the last to Galois actions on $\ell$-adic vector spaces. However, Galois actions are used in different ways in the three. Thinking about the three concepts, you might be led to questions like these:
Can we construct Galois representations from the Galois action on the fundamental group of a curve? (this would be the first step in relating motives to anabelian geometry)
Do these Galois representations arise from motives? (this would be the second step)
Are these motives related to the geometry of a curve? (seeking a deeper connection to anabelian geometry)
Can the class of motives arising this way be used to construct or describe the motivic Galois group? (now we bring in Grothendieck-Teichmuller theory)
I think these questions at least touch on the beginning of what Grothendieck was thinking of.
Since Grothendieck, people have heavily studied these questions, primarily in the case of unipotent quotients of the fundamental group, starting with the paper of Deligne on the fundamental group of the projective line minus three points. I think it's fair to say that the answer to all these questions is yes, with the largest caveat for the last question - I believe we can understand certain very special quotients of the motivic Galois group this way, but I don't think anyone has a strategy to construct the whole thing.
The story goes something like this:
Deligne looked at the maximal pro-$\ell$ quotient of the geometric fundamental group of the projective line minus three points. This is naturally an $\ell$-adic analytic group, and has a Lie algebra, which is an $\ell$-adic representation, and admits an action of the Galois group. This is supposed to be the $\ell$-adic realization of a motive, and Deligne worked to find the other realizations, including Hodge theory. This is a mixed motive, not a pure motive, so isn't constructed directly from linearizing algebraic varieties.
All motives generated this way are mixed Tate motives, i.e extensions of powers of the Tate motive (the inverse of the Lefschetz motive), and are everywhere unramified. One can define the Tannakian category of everywhere unramified mixed Tate motives, and the Tannakian fundamental group is known to ask faithfully on the limit of these unipotent completions.
Best Answer
Check out Section 2 of Noohi's paper Fundamental groups of topological stacks with slice property, Algebr. Geom. Topol. 8 (2008) pp 1333–1370, doi:10.2140/agt.2008.8.1333, arXiv:0710.2615.