A comparison between the fundamental groupoid and the fundamental group

One example would be with $X$ and $Y$ indiscrete spaces of cardinalities $1$ and $2$ respectively. Their fundamental groupoids have different cardinalities and therefore are not isomorphic, but both of their fundamental groups are trivial.

It suffices to prove the following:

If $X$ and $Y$ are connected groupoids, then given any bijection $F : \operatorname{ob} X \to \operatorname{ob} Y$ and any group isomorphism $\Phi : X(x, x) \to Y(y, y)$ (where $x$ is in $X$ and $y = F x$), there is a groupoid isomorphism $X \to Y$ extending the given data.

Choose an isomorphism $p_{x'} : x \to x'$ for each $x'$ in $X$, with $p_x = \mathrm{id}_x$, and choose an isomorphism $q_{y'} : y \to y'$ for each $y'$ in $Y$, with $q_y = \mathrm{id}_y$. We define a functor $F : X \to Y$ as follows: $F$ acts on objects as the given bijection $\operatorname{ob} X \to \operatorname{ob} Y$, and for each morphism $f : x' \to x''$ in $X$, we define $$F f = q_{y''} \circ \Phi (p_{x''}^{-1} \circ f \circ p_{x'}) \circ q_{y'}^{-1}$$ It is easily shown that $F$ is indeed a functor and has the desired properties.