Short answer: as stated, Ω is not a linear order, and it is far too complicated to admit a nice “universal” combinatorial description. In fact, it already contains, as a suborder, the very complicated poset (P(ω), ⊆*) of subsets of the natural numbers ordered by inclusion modulo finite. So any hope for a simple colimit description in the spirit of your Θn- or Φ-examples cannot work without imposing strong regularity restrictions on the functions.

Details:

1. Your relation is just Big-O. For real-valued functions f,g defined eventually (say on (λ,∞)), your definition f ≼ g ⇔ max{|liminf f/g|, |limsup f/g|} < ∞ is equivalent to limsup |f/g| < ∞, i.e., f = O(g) as x → ∞. The equivalence relation f ≃ g means f = Θ(g) (mutual O).

2. Ω is not linearly ordered. Even among continuous functions you get incomparability: take f(x) = sin x and g(x) = cos x. Then limsup |f/g| = limsup |tan x| = ∞ and limsup |g/f| = ∞, so neither f ≼ g nor g ≼ f. Thus Ω/≃ is only a partial order.

3. Ω contains (P(ω), ⊆*) as a suborder. This shows it has antichains of size 2^ℵ0 and in general is very complicated.

   Construction. Fix a nonnegative C∞ bump b supported in [0,1] with max b = 1, and let bn(x) = b(x−n). Choose a rapidly increasing sequence cn → ∞ (e.g. cn = 2^n). For A ⊆ ω, define fA(x) := 1 + Σ_{n∈A} cn bn(x). Then fA is continuous on [0,∞). The supports of the bn are disjoint, so at any x at most one term contributes.

   Claim. fA ≼ fB iff A ⊆* B (i.e. A\B is finite).

   • If A ⊄* B, then there are infinitely many n ∈ A\B. At points xn with bn(xn) = 1 we have fA(xn) ≥ cn+1 while fB(xn) ≤ 1, hence fA(xn)/fB(xn) ≥ cn+1 → ∞. So fA ≰ fB.
   • If A ⊆* B, choose N so that A\B ⊆ {0,1,…,N−1}. For x ≥ N, three cases can occur: either no bump is active (fA = fB = 1), or n ∈ B\A is active (fA = 1, fB ≥ 1), or n ∈ A∩B is active (fA = fB = 1 + cn bn(x)). In all cases fA/fB ≤ 1 for x ≥ N, so limsupx→∞ fA/fB ≤ 1 < ∞, i.e. fA ≼ fB.

   Thus A ↦ [fA] is an order-embedding of (P(ω), ⊆*) into Ω/≃.

Consequences:

• Ω/≃ has antichains of size continuum and embeds every countable partial order. In particular, it is not a linear order and admits no simple “lexicographic” description.
• Your nice chains Θn, Φ, Ψ sit inside Ω/≃ as very special, highly regular suborders (they live in Hardy-type contexts), but they are a tiny, tame part of the whole poset.

4. If you want a linear order, restrict the class of functions. A standard way is to work with germs at +∞ of eventually positive, eventually monotone functions from a Hardy field (or more generally definable in an o-minimal structure). On such classes, comparability by O is a total order and is well understood:

• For logarithmic–exponential (LE) functions one gets a lexicographic hierarchy of “monomials” akin to your Θn, Φ, Ψ and beyond (transseries). See Rosenlicht; van den Dries–Macintyre–Marker (1994) on LE-series; van den Dries–Miller on Hardy fields.
• These ordered structures embed into the Hahn field of transseries and further into the surreal numbers, yielding very nice linear orders.

But without such regularity assumptions, Ω/≃ is as wild as (P(ω), ⊆*), so there is no simple combinatorial or universal colimit-type description of its isomorphism type.