Comparison in an Elliptic Curve group
If we specialize a generator $G$ of an Elliptic Curve group $(\mathbb G,+)$, we can define a function $f_G:[0,n)\to\mathbb G$ where $n$ is the group order, as (with $\infty$ the group neutral/point at infinity)$$f_G(a)=\begin{cases}\infty&\text{if }a=0\\f_G(a-1)+P&\text{otherwise}\end{cases}$$
In other words, we defined $f_G$ such that $f_G(a)=a\cdot G$ where $\cdot$ is "scalar multiplication". This $f_G$ is a bijection, which inverse function $f_G^{-1}$ is well defined, even if $f_G^{-1}(P)$ is hard to compute for random given $P$. Thus the natural comparison in $[0,n)$ yields one comparison $\mathbb G$ (dependent on $G$) for elements of the Elliptic Curve group, defined as $P\;<_GQ\iff f_G^{-1}(P)<f_G^{-1}(Q)$.
Considering this why would having comparison break ECC?
Having a computable comparison breaks ECC. Given $G$ and an oracle that given $P$ and $Q$ tells if $P\;<_GQ$, we can compute the Discrete Logarithm to base $G$ (which breaks ECC) with less than $\log_2 n$ queries, by finding $x$ such that $x=f_G^{-1}(P)$ by binary search.
Modulo in Elliptic Curve Groups
For any generator $G$, it holds $P+Q=f_G(f_G^{-1}(P)+f_G^{-1}(Q)\bmod n)$. We can similarly define $P\ *_G Q$ as $f_G(f_G^{-1}(P)*f_G^{-1}(Q)\bmod n)$. Now $(\mathbb G,+,*_G)$ is isomorphic to the finite ring $\mathbb Z/n\mathbb Z$ of integers modulo $n$. With a stretch of imagination, any (of a few) modulo operators we can define in $\mathbb Z/n\mathbb Z$ yields a modulo (dependent on $G$) for $\mathbb G$.
What modulo if (computable) efficiently would break ECC & how?
Define modulo in $[0,n)$ the usual way, that is$$u\bmod v=w\iff0\le w<v\text{ and }\exists k\in\mathbb Z\text{ such that }u=k\,v+w$$
For a fixed generator $G$, that defines a modulo $\bmod_G$ in $\mathbb G$ as above. For $Q\ne\infty$ and any $P\in\mathbb G$, it holds $(P\,\bmod_G Q)=P\,\iff\,P\;<_GQ$. Thus ability to compute $\bmod_G$ implies ability to evaluate $<_G$, and thus break ECC in the group as seen before. There are actually better methods (requiring considerably less queries to a $\bmod_G$ oracle) by using the $\bmod_G$ to compute a Greatest Common Divisor.
Generalization to Finite Cyclic Groups
By definition they have a generator. It can be used to construct an isomorphism to the ring $\mathbb Z/n\mathbb Z$, and the same reasoning holds.
