abbreviations - Usage of the word "one" in conjuction with name in genetive, like this example

Your first option is very awkward and, I would say, grammatically incorrect. If anything, it should read

...the one of Aladdin.

or

...it's one of Aladdin's carpets.

If you want to maintain the possessive, I suggest

It's Aladdin's carpet

The plural "carpets" is not necessary—the phrase "Aladdin's carpet" is almost like a collective noun. It refers to a single carpet, but it doesn't matter which one.

The "should be" in your response is what is making constructing the rest of the sentence difficult. If your friend is asking for an explanation of the comic, I would say

The reference is to Aladdin's carpet.

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