Congruency works just as well in elliptic and spherical geometries, and it also works just as well in any higher dimensions, $2$ or more. Congruency can be used to define every geometric relation, including various notions of betweenness, separation, colinearity, and the order-type for lengths.
The situation with spherical geometry is more complicated, but it's trivially easy to obtain the desired axiomatization for elliptic geometry. After defining the primitive notions related to lines and circles, we can simply take Euclid's axioms with the appropriately modified parallel postulate. We'll also need a few other minor axioms to fill gaps in Euclid's reasoning, and lastly, we just include an axiom schema of completeness for line segments, same as Tarski. That's it!
Expressiveness
Two geodesic segments (line segments) are congruent if and only if they have the same length. Given points $x,y$, there is at least one minimal-length path connecting $x$ to $y$, and due to the previous observation, this path is unique modulo congruence. Therefore, the relation $xy\equiv zw$ can be interpreted as saying that some minimal-length path from $x$ to $y$ is congruent to some minimal-length path from $z$ to $w$. This is exactly equivalent to asserting $d(x,y)=d(z,w)$, under the appropriate distance metric, so congruence has an unambiguous meaning in elliptic geometry.
To see how congruence can be used define other basic properties about lines and circles, consider the following definition of the distance order relation. This characterization is more well known in the context of flat and hyperbolic geometries, but it works just as well in elliptic and spherical.
$$xy\leq zw \iff \forall u, (xu\equiv yu)\implies \exists v, (zv\equiv wv \equiv xu)$$
To understand this, consider the set $D_{x,y}=\{d(x,u) : xu\equiv uy\}$, and simply notice that $d(x,y)\leq d(z,w) \iff D_{x,y}\supseteq D_{z,w}$. This works since $D_{x,y}$ is always an interval subset of $\mathbb{R}^+$, the minimum of $D_{x,y}$ is strictly increasing with $d(x,y)$, and the maximum is non-increasing. In particular, the minimum of $D_{x,y}$ is always exactly $d(x,y)/2$, obtained by taking $u$ to be a midpoint of any minimal path from $x$ to $y$. The supremum of $D_{x,y}$ depends on the geometry, but in spherical it's $\pi-d(x,y)/2$, obtained where $u$ is antipodal to the midpoint of $x,y$. In Elliptic the maximum is instead $\frac{\pi}{2}$, obtained when $u$ is mutually orthogonal to both $x$ and $y$. This allows us to define two related notions of betweenness, distinct only in the compact geometries.
$$\begin{align}
B!xyz &\iff \forall u, (xu\leq xy \land uz\leq yz)\implies (y=u) \\
Bxyz &\iff \forall u, (xu\leq xy \land uz\leq yz)\implies (xu\equiv xy \land uz\equiv yz)
\end{align}$$
The weaker relation $Bxyz$ conveys that $y$ exists on some minimal path from $x$ to $z$. The stronger relation $B!xyz$ means that $y$ exists on every possible minimal path. It turns out that, so long as $x,z$ are not maximally distant, there is always a unique minimal path between them. In such cases we get $\forall y, Bxyz\iff B!xyz$, which is always the case in flat and hyperbolic geometries. Similarly, so long as $x\neq y$ and $x,y$ are not maximally distant, there will be a unique ray starting from $x$ and crossing $y$, and likewise a unique line. These notions are definable as follows.
$$\begin{align}
Rxyz &\iff (Bxyz \lor Bxzy) \\
\overline{R}xyz &\iff \exists(\epsilon,\delta\neq x), Rxy\epsilon \land B\delta x\epsilon \land Rx\delta z \\
Lxyz &\iff Rxyz \lor \overline{R}xyz
\end{align}$$
The relation $Rxyz$ expresses that $z$ lay on some ray which starts from $x$ and crosses $y$. In compact geometries like spherical and elliptical, the ray is considered to terminate once it reaches a point maximally distant from $x$. This allows us to distinguish between directions, even if there are multiple geodesics between any two points. The relation $\overline{R}xyz$ defines a sort of anti-ray, another ray that's likewise rooted at $x$ but which extends in the exact opposite direction. By taking the union of both the ray and antiray, the relation $Lxyz$ expresses colinearity. You can check the edge cases (e.g. when $x=y$) to verify that $L$ still behaves as you'd expect. The separation relation, mentioned in the comments, could be defined like so.
$$\begin{align}
Lxyzw &\iff (Lxyz \land Lxyw \land Lxzw \land Lyzw) \\
xySzw &\iff Lxyzw \land (Bxzy\lor Bxwy) \land (Bzxw \lor Bzyw)
\end{align}$$
Completeness
To convince yourself that we can define every geometric relation, it suffices to construct a model of flat geometry, inside which we construct a model of elliptic geometry, and finally we need to construct an isomorphism between the metatheory's elliptic universe and the inner elliptic model. In this answer to another question, I outline how we can construct the required models and isomorphisms, and the situation is roughly the same for either elliptic or spherical. Therefore, we only need to prove the following facts.
- Our primitive notions are expressive enough to define those models and isomorphism,
- We prove all of Tarski's axioms about our model of flat geometry, and
- We prove the previously described isomorphism is actually an isomorphism.
The first item is satisfied so long as we can define the basic properties about line segments and circular arcs, which we've already demonstrated. The second and third items are more tedious, but only rely on basic facts about lines and circles, which should be included in our axiomatization. For the second item, obtaining Tarski's completeness axiom about the flat model follows almost immediately from the completeness of line segments in the metatheory.
Because we'll have all of Tarski's axioms about our model of flat geometry, then our theory will prove every true statement about flat geometry. It follows that we also prove every true statement about the inner model of elliptic geometry. Since we prove isomorphism between the elliptic model and the elliptic universe, we therefore prove every true statement about the elliptic universe. Similarly, we obtain every geometrically definable relation/function by passing through the isomorphism. In other words, our theory will be complete, decidable, and maximally expressive (relative to Tarski's geometry).
