Modifying nouns is the definition of and natural role of an adjective, e.g., 「彼は僕と同じ近所で住んでいます」⇒"He lives in the same neighborhood as me."
The strange thing (compared to English) in the original sentence is, I think, the way that 同じ takes a subject while still acting as an adjective. That's just something that the Japanese language can do that English can't. い-adjectives are basically verbs (some textbooks even call them "descriptive verbs" instead of calling them adjectives), and 同じ is basically an い-adjective that just happens to be missing the い on the end (probably a historical accident of the sound getting shortened).
In any case, I would not say a copula has been omitted, but rather that in an adjectival sentence a copula is added for grammatical completeness. Sort of.
同じ is a weird case.
With regular い-adjectives the copula is not added after them, and with な-adjectives, they have the な form of the copula when modifying a noun. 同じ is peculiar in acting like an い-adjective in most ways but taking the copula when at the end of a sentence. (It's arguably a noun when you do that, but then again, it's arguable that all な- and の-adjectives are just descriptive nouns.)
Anyway, in your original sentence 同じ was not acting in any way unique or peculiar, it was just being an adjective. You could similarly have,
「とある髪が赤い男が駅の前で待っていた」⇒"A certain red-haired man was waiting in front of the station"
In this case, 彼は髪が赤い would not take a copula (it might take です for politeness, but not as a copula, and not だ at all), so hopefully that clarifies why I wouldn't consider there to be any omitted copula.