The fundamental groupoid $\Pi_1(M)^{top}$, when equipped with the 'usual topology' when the space $M$ permits it, is weakly/Morita equivalent to the same groupoid equipped with the discrete topology, call it $\Pi_1(M)^\delta$. This is essentially Proposition 4.42 in my thesis. Even better, the identity functor $\Pi_1(M)^\delta \to \Pi_1(M)^{top}$ is a weak equivalence. If a functor $X\to Y$ of topological groupoids is a weak equivalence, then $\Pi_1(X) \to \Pi_1(Y)$ is an equivalence of groupoids. And lastly, for a topological groupoid $X$ that has the discrete topology, the canonical functor $X\to \Pi_1(X)$ is an equivalence. So for a suitable topological space $M$, there is a canonical functor $Pi_1(X)^\delta \to \Pi_1(\Pi_1(M)^{top})$ and it is an equivalence.

The fundamental groupoid $\Pi_1(M)^{top}$, when equipped with the 'usual topology' when the space $M$ permits it, is weakly/Morita equivalent to the same groupoid equipped with the discrete topology, call it $\Pi_1(M)^\delta$. This is essentially Proposition 4.42 in my thesis. Even better, the identity functor $\Pi_1(M)^\delta \to \Pi_1(M)^{top}$ is a weak equivalence. If a functor $X\to Y$ of topological groupoids is a weak equivalence, then $\Pi_1(X) \to \Pi_1(Y)$ is an equivalence of groupoids. And lastly, for a topological groupoid $X$ that has the discrete topology, the canonical functor $X\to \Pi_1(X)$ is an equivalence. So for a suitable topological space $M$, there is a canonical functor $\Pi_1(X)^\delta \to \Pi_1(\Pi_1(M)^{top})$ and it is an equivalence.