This answers address the part of the question asking what the divisors above correspond to in number theory and ring theory. I also include some vague remarks on the naming.
The relevance of Dedekind domains in the geometric context was already mentioned in a comment. And, the notion of Dedekind domains in some sense 'comes from' number theory.
The rings of algebraic integers of algebraic number fields are important examples of Dedekind domains and the historical root for the development.
Rings of algebraic integers are not necessarily unique factorization domains, and one tried to remedy this lack of uniqueness of factorization into irreducible elements by passing to a different (larger) structure. A structure consisting of 'ideal elements' that would not lack the nice property of having unique factorizations into irreducibles.
Now, todays ideals are in some sense these 'ideal elements' for rings of algebraic integers and more generally Dedekind domains.
(One way to define the notion Dedekind domain is: every non-zero ideal is a product of prime ideals.)
Yet, besides the approach via ideals there was also a different/competing approach, namely that of establishing a divisor theory (orig. Divisorentheorie) promoted by Kronecker.
(When I searched for a reference to link to, almost first thing I found was a related MO question, how practical!)
In the number theoretic context today this approach and name is not very wide spread, but might explain the name. And, it is still preserved in more general contexts (cf. below).
But, it is essentially equivalent to considering ideals. And, as mentioned in one of the comments (in suitable circumstances) the fractional ideals correspond to the divisors (in the geometric context).
As said above for a Dedekind domain all non-zero ideals are in a unique way products of prime ideals, so $I= \prod_P P^{v_p}$ where $P$ runs through the non-zero prime ideals and $v_p$ are nonnegative integers all but finitely many $0$. If one allows $v_p$ to be negative too, one gets the fractional ideals.
The link is perhaps even better seen when generalizing a bit, and here the standard terminology even includes words resembling 'divisor'.
Another way to define Dedekind domain would be to say it is a noetherian, integrally closed domain of dimension $1$.
Now, here is a generalization of Dedekind domains: Krull domain.
A way to define what a Krull domain is, is to say that it is a completely integrally closed domain and a v-noetherian domain; where v-noetherian means ACC only on divisorial ideals that is those for which $(I^{-1})^{-1} = I$, in particular v-noetherian is weaker than noetherian; and where completely integerally closed is a condition that is in principle stronger than integrally closed but for noetherian domains it is the same.
So, since Dedekind domains are noetherian and integrally closed they are completely integrally closed, and since they are noetherian they are v-noetherian. Thus, they are Krull domains.
More generally, every integrally closed and noetherian domain is Krull (without a condition on dimension!).
In particular, Krull domains have the property that the localization of the domain at every height one prime ideal is a DVR, and with an extra condition one could take this property as definition, too.
Now, the problem arises that also in Krull domains one would like to have 'ideal elements' or a 'divisor theory'. Yet, the usual ring ideals are in this case not the 'ideal elements'. One needs to consider a different type of ideals namely the divisorial ideals.
It is then true that every divisorial ideal can be uniquely factorized as a product of prime divisorial ideals (in perhaps more classical terms these are the prime ideals of height one). A key-point is that principal ideals are divisorial ideals, and what one cares about mainly is to have a substitute of unique factorization for the ring elements, so principal ideals.
The product used for this factorization is not the usual product of ideals, but rather the product as divisorial ideals that is one forms $((IJ)^{-1})^{-1}$ to get the product of $I$ and $J$.
One can still generalize this more to commutative semigroups with identity and cancellation law. Namely, a semigroup $S$ with a divisor theory is one for which there exists a free commutative semigroup $F$ and a homomorphism $f:S \to F$ such that $a|b$ if and only if $f(a)|f(b)$ for all $a,b \in S$ [one implication is trivial as $f$ is a homomorphism, but the other one is important]. In other words, the arithmetic (or divisibility relations in $S$) are governed directly by the ones in a free semigroup (so a factorial/unique factorization one).
Side note: the $f$ above is not strictly speaking a divisor theory as there is a minimality condition omitted, but still one can rigorously take this as definition; let us assume for simplicity we have in addition this suitable minimality condition (whose definition we omit).
Then we have an $S$ which we care about and an $F$ which is easy to understand (consisting of the 'divisors'). So, the divisors of $S$ control or govern the divisibility relations in $S$. To be a bit more precise, in $F$ itself we only have what corresponds to the positive divisors (that is no negative 'coefficient'). However, to make the link more direct, one can (and does, e.g., for defining the class group) instead consider the quotient group, which consists precisely of all (formal) products $\prod_p p^{s_p}$ where $p$ runs through the prime elements of $F$ the $s_p$ are integers (all but finitely many $0$). [The sole difference is that here multiplicative instead of additive notation is used.]
What is the link of these semigroups to domains: a domain is Krull if and only if its multiplicative semigroup is a semigroup with divisor theory.
In fact, one can take the semigroup of fractional ideals for the $F$.
And, since a Dedekind domain is Krull its multiplicative monoid is also a semigroup with divisor theory and in this case one can take $F$ to be the (usual) ideals.
In view of the above, if one completely reduces down to the most classical number theory situation of the integers or natural numbers then everything 'collapses' and the positive divisors (so no negative 'coefficient') 'are' just the elements of the original structure, except for zero and 'forgetting the sign' for the integers. [Omitting the positivity condition, the totality of divisors would correspond to the respective quotient group, i.e., the positive rationals with multiplication (one allows positive and negative exponents in the 'prime factorization')]
In the classical situation of algebraic number theory (ring of integers of an algebraic number field) and more generally Dedekind domains the positive divisors correspond to the ideals and the divisors to the fractional ideals.
And, one could say, these divisors control or govern the divisibility relations (in an efficient way). This is also true for the integers; what one omits is the sign which is in fact irrelevant for divisibility.
So, the divisors (in the geometric context) are closely linked (in certain cases the same) as objects that in number and ring theoretic contexts describe or govern the arithmetic, i.e. divisibility relations.
