Consider the following two sentences.
A: 僕は左と右が区別できない。
B: 僕は左と右を区別することができない。
I often see B but just know A now.
How can A be possible? What kind of grammar is A?
The following sentence (from the Tanaka Corpus) inspired me to ask this question:
その[坊や]{ぼうや}はツバメとスズメが[区別]{くべつ}できない。
The boy can't tell a swallow from a sparrow.
Both are correct. できる means exactly the same thing as することができる. You can treat it as a special potential form of する.
So it's not that できる is being attached to a noun, it's that it is taking the place of する in a する verb.
Here are a couple more examples:
この部屋はうるさくて勉強できない。（勉強することができない）
明日の予約が確認できた。（予約を確認することができた）
Note that it's most common to mark the thing that you can do with が rather than を because, like potential forms in general, できる really means that such-and-such "is doable." You'll see を sometimes but you can consider が the the traditionally correct way to phrase it.
Hope that helps!
