TL;DR The higher étale homotopy groups are the homotopy groups of the profinite completion of the shape of the étale topos. As such they are profinite groups. If you choose to see profinite groups as topological groups, group schemes or pro-systems is largely a matter of choice.

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How is the étale homotopy group defined? There are many ways, but possibly the most elegant understanding of it, can be obtained using the theory of the [shape](https://ncatlab.org/nlab/show/shape+of+an+%28infinity%2C1%29-topos) of an $∞$-topos. Under this theory, for any ($∞$-)topos $\mathscr{X}$ we can obtain a functor
$$\mathrm{sh}(\mathscr{X}):\mathrm{Space}→\mathrm{Space}\,,$$
where $\mathrm{Space}$ is the $\infty$-category of spaces, by sending a Kan complex $K$ to $\Gamma(\mathscr{X},\underline{K})$, the (derived) global sections of the constant sheaf at $K$ [1]. Since this functor commutes with finite (homotopy) limits, it is of the form
$$ \mathrm{sh}(\mathscr{X})(K)=\mathrm{colim}_i\mathrm{Map}(U_i,K)$$
for some (uniquely determined) pro-system of spaces $\{U_i\}_i$.

Once we have the pro-space $\mathrm{sh}(\mathscr{X})$, we can consider its profinite completion, which in this language is simply the restriction of the functor to the ($∞$-)category of $\pi$-finite spaces (those spaces that have only finitely many nonzero homotopy groups, and such that all those homotopy groups are finite). This has also the nice technical advantage of ignoring the difference between a topos and its hypercompletion. This corresponds to replacing every $U_i$ in the above pro-system by "$\pi$-finite approximations" (i.e. by the pro-system of $\pi$-finite spaces with a map from $U_i$).

Long story short, we can associate to the étale topos of a scheme $X$ a (uniquely determined) pro-system $\{X_i\}_i$ of $\pi$-finite spaces. Moreover, since this construction is functorial in the scheme, to every geometric point $x:\mathrm{Spec}\,\Omega\to X$ we can associate a map of pro-systems $\ast→ \{X_i\}_i$, that is up to reindexing, a coherent choice of basepoint $x_i$ to each $X_i$. In particular, for any $n\ge 1$, we can consider the pro-system of groups
$$ \{\pi_n(X_i,x_i)\}_i$$
Since all $X_i$ are $\pi$-finite, this is in fact a pro-system of finite groups. But pro-systems of finite groups are well known to be exactly the same thing as profinite groups.

In particular you could see $\pi_n(\mathscr{X},x)$ as some kind of topological group (since we know that profinite groups are the same thing as compact totally disconnected topological groups), although how useful that is in practice I couldn't say. You could also see it as a particular group scheme (and this can be useful to compare it to other constructions that naturally produce a group scheme).

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To see that this definition recovers the classical étale fundamental group when $n=1$, you need a theorem telling you that finite étale covers are equivalent to finite locally constant sheaves of sets over the étale site. Then it's just a matter of chasing universal properties along the various constructions.

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[1] Note that when $\mathscr{X}$ is the topos of a paracompact Hausdorff space $X$, $\Gamma(\mathscr{X},\underline{K})$ is just the space of maps from $X$ to $K$. So we can imagine this functor as a substitute for the (nonexistent) functor $\mathrm{Map}(\mathscr{X},-)$.