In ancient times divisibility was conceptualized by "measurement". Thus the ancients would write $\,n\,$ measures $\,a\,$ for our $\,n\,$ divides $\,a,\,$ and the divisor was thought of as a "unit of measurement" - which is one of the meanings of "modulus" in Latin. Therefore, expressed in ancient terms, $\,a\equiv b\pmod n\,$ was viewed geometrically as: the distance between $\,a\,$ and $\,b\,$ can be measured exactly using a ruler whose unit of measurement ("modulus") $= n$.
In modern times the denotation has been greatly generalized so that roughly speaking $\, x\bmod y\,$ simply means $\,x\,$ assuming $\,y\,$ [often $\,y\,$ is viewed as "small" in some sense, e.g. the proof is complete (mod trivial edge cases left to the reader)].
When $\,y\,$ is an equality (or set of such) then $\,x\bmod y\,$ means we are working modulo the smallest equivalence relation generated by the equalities that is additionally compatible with any ambient structure, e.g. in an algebraic structure it denotes the congruence relation generated by the equalities (said equivalently, assuming all the equalities that can be derived from the listed equalities using the axioms of the algebraic structure).
In algebraic structures (like rings & groups) enjoying "subtraction" we can "zero-normalize" all of our equations $\,a = b \iff a-b = 0,\,$ so we can write $\,x\bmod y,z\,$ for $\,x\bmod y=0,z=0.\,$ Here congruences are completely determined by the equivalence class of $\,0\,$ (so-called ideal-determined algebras), so congruences can be replaced by ideals, and instead of writing equations we can simply list the generators of the ideal, e.g. $\,R[x]\bmod (p,x^2+1)$ denotes the quotient ring $R[x]/(p,x^2+1),\,$ which is the most general ring image of $R[x]$ assuming $\,p=0\ \ \&\ \ x^2+1 = 0.\,$ Similarly for groups, e.g. in presentations given by generators and relations, the relator elements $\,r,s\,$ in $\,\langle x,y\mid r,s\rangle\,$ denote the equations $\,r=1,\,s=1\,$ in multiplicative notation.
The term "modular design" is somewhat related, since decomposing a system into related component modules can be viewed as a (non-mathematical) generalization of the above said partition into equivalence classes that is induced when working mod equations. Both methods can be viewed as instances of the divide and conquer method of problem solving. Of course in algebra we generally have much more innate structure that we can exploit.
For example, decompositions into products of "simpler" structures is a prototypical algebraic way of dividing and conquering in algebra. This is familiar elementwise in terms of integer factorizations into products of primes - then working "locally" on each prime(power) factor (or more general local-global methods). More structurally we can divide using ring factorizations given by CRT = Chinese Remainder Theorem. More broadly there are such product decompositions for general algebras, e.g. by Birkhoff's Theorem every algebra $A$ is isomorphic to a subalgebra of a sd(subdirect)-product of sd-irreducible images of $A,$ so we reduce to studying sd-irreducible algebras (this generalizes Stone's representation theorem for Boolean algebras, where the only sd-irreducible Boolean algebra is the two-element Boolean algebra; similarly for vector spaces (over a fixed field), and a one-dimensional vector space).
