We use the notation $\mathbb N^{\gt 0} = \{1,2,\dots,n,\dots\}$.
If $m,n \in \mathbb N^{\gt 0}$ we can always apply Euclidean division to get a quotient - if $m \ge n$ we can call $m$ the dividend and $n$ the divisor, and if $m \lt n$ we can call $n$ the dividend and $m$ the divisor. This is a commutative binary operation, $\mathsf {EC}(m,n)$.
Examples: $\mathsf {EC}(3,5) = 1$,$\;\mathsf {EC}(11,11) = 1$ and $\mathsf {EC}(2,7) = 3$.
A mapping $f: \mathbb N^{\gt 0} \to \mathbb N^{\gt 0} $ is said to have $+\infty$ as a limit if for every $M \in \mathbb N^{\gt 0}$ there exist $N \in \mathbb N^{\gt 0}$ such that for every $n \ge N$ the image $f(n)$ is greater than or equal to $M$.
Let $f$ and $g$ both have $+\infty$ as a limit. We can define other mappings for each $k \in \mathbb N^{\gt 0}$,
$\tag 1 k \times \mathsf {EC}(f,g): \; n \mapsto \text{Max}[\;\mathsf {EC}(kf(n),g(n)),\,\mathsf {EC}(f(n),kg(n))\;]$
Definition: Two mappings $f$ and $g$ are said to approach $+\infty$ at the same rate if for every $k \in \mathbb N^{\gt 0}$, $k \times \mathsf {EC}(f,g)$ is eventually constant and equal to $k$.
Example: $f(n) = n^2 + 100n + 10000$ and $g(n) = n^2$ approach $+\infty$ at the same rate.
Question 1: Did this concept appear in Greek_mathematics or, for that matter, have these definitions ever been used in the mathematical literature?
I find it interesting that we can get the concept of a limit at a foundational level - our universe of discourse is restricted to only the natural numbers.
Question 2: Are there known results in mathematics that can be developed and expressed (perhaps in a watered down fashion) from this primitive platform?
Yes, this is a very important and well-understood notion.
Your definition is more concisely written as $$\lim_{n\rightarrow\infty}{f(n)\over g(n)}=1,$$ which in turn is generally abbreviated "$f\sim g$." Here I'm taking the limit over naturals, so $f$ and $g$ are understood as functions with domain $\mathbb{N}$; we can just as easily define $\sim$ for functions on the rationals, reals, complexes, or etc. Some basic properties are described here, and you may also be interested in other varieties of asymptotic comparison.
As to its history, I'm unaware of any treatment of growth rates in "ancient" mathematics, Greek or otherwise - so far as I know, the earliest such investigations occurred in the $1800$s in the context of analytic number theory (e.g. the prime number theorem is a result in asymptotic analysis) - but it's hard to prove a negative, and I could very easily be wrong.