I was wondering if we can use the word "luxury" to refer to a "luxurious item",
For example, are the sentences below considered grammatical? :
I have a luxury.
I have one luxury.
I have three luxuries.
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Luxury is an abstract object, but it can be countable, and you can use it to refer to a certain luxurious item. All the sentences you have in the question are grammatical. You can say "the divan is a luxury", or "I have the luxury of a Corvette" even "I have a luxury" (properly explained with context); but luxury by itself doesn't mean anything specifically physical.
I must disagree with the comment implying luxury can't be used as in OP's three examples. This NGram for "one luxury is" returns nearly 3000 results, most of which treat a "luxury" as a countable noun.
I don't much like OP's first example, but we don't really have much context, and I certainly don't dispute it on grammatical grounds. The others seem unremarkable to me - here are many written instances of "had two luxuries", for example.
One luxury I allow myself on ELU is disputing unsubstantiable statements. Another is using words like unsubstantiable (incapable of being substantiated), despite my browser spell-checker objecting!