Of course chicken describes a property of the soup, just as lentil or cabbage do in lentil soup or cabbage soup. Chicken is an attributive noun (or noun modifier) in the phrase chicken soup. It tells us what is in the soup. We would not say "I need a chicken one" in answer to your query. Instead, we would say: "I need the chicken." Why? The best answer is simply "Because that is how English is spoken."

If chicken does not modify soup, you need to try different soups.

@oerkelens - Ha ha, right on. Clearly, this is chicken soup, and this isn't. How can chicken not be describing a property of the soup? (Note to user178049: The comment is meant to say that, if the chicken doesn't modify the soup, then the soup is too watery and not well-made. As a comment, it's both humorous and insightful.)

@P.E.Dant I hope you could post it as an answer.

There are already several answers here. Search this site for noun adjunct or attributive noun.

Possible duplicate of is "release date" grammatically correct?

NO @P.E.Dant, this is not a duplicate. I know what a noun adjunct is. I'm trying to distinguish between a modifier/adjunct and complement.

What is the difference between "specifying what kind of soup it is" and "describing the soup"? How can you do one without doing the other?

@P.E.Dant This is really hard to account for. But I could give you another example. In an NP like 'A hotel manager', the hotel doesn't describe the manager; rather, it tells you what is managed.

No, it's not a complement. "Chicken" is a nominal modifying "soup". Of course it describes the soup: It's chicken-flavoured soup.

There is no need. Read snailplane's comprehensive answer to the duplicate question, which is here. Mine would be found wanting in some wise, I fear.

@P.E.Dant Snail's answer is indeed very good! And it answers my question in the title. But it doesn't solve my problem, which is regarding the difference between a complement and a modifier.

Complements: "a linguistics student" (a student of linguistics), "a flower seller" (a seller of flowers), "a legal advisor" (an advisor on legal matters) and so on. Can you see how they differ from modifiers?

@BillJ It's easy to see how one might think, then: "chicken soup" (soup made of chicken). I'm beginning to understand the OP's dilemma here, but hanged if I can resolve it easily.

The question's title might better be: Noun adjunct or complement?

Normally complements have to be licensed by a head word, but not with pre-head complements in NPs. The crucial point is that they have paraphrases involving post-head complements where a forced choice of preposition is observable, e.g. the examples I gave. Also, compare an alcohol ban and a ban on alcohol; a wrist injury and an injury to the wrist.

@BillJ I see that Huddleston has "Complements express semantic arguments of the head noun—property, relation, process, or action. He's a legal advisor. [He's a professional in the field of law.]" There's a basis for a good answer there, maybe: chicken is not a property of but a component of the soup.

@BillJ I can see it differs. But it's not a clear-cut explanation to me. I'll let you know if I have some troubles.

Perhaps @StoneyB will notice this cascade and come up with a good explanation of the distinction between complement and noun adjunct in chicken soup. It seems to me that there is a case to be made for both determinations. Surely a soup of chicken provides the choice of preposition.

@P.E.Dant I agree with BillJ here that chicken is more complement than modifier. CGEL's test for complementarity with nouns not derived from verbs (p.441) is whether replacing the head noun with the pronoun one results in some "degree of infelicity": how do you feel about John made an onion soup, and Jack a chicken one?

@StoneyB From one end of the "internet" to the other, chicken soup is held aloft as an exemplar of the attributive noun or noun adjunct! Someone has a metric buttload of editing in store. And see the OP's "I need a chicken one". Doesn't that fail the CGEL test, and miserably? I am no nearer clarity.

@P.E.Dant OP's "I need..." is marked (correctly) as unacceptable, with a *. ... "Attributive" these days mostly designates the position rather than the function, and "adjunct" as far as I can tell doesn't actually mean much of anything; they're both hangovers (like 'gerund' and 'participle') from older grammatical understandings. What we've really got a crying need for is an English grammar translated into the English language.

OK, I'm officially at sea now.

chicken soup: what is the chicken?

from BillJ: "a linguistics student" (a student of linguistics), "a flower seller" (a seller of flowers), "a legal advisor" (an advisor on legal matters) ... to which add "chicken soup" (a soup of chicken). And then: a student, a seller, an advisor are all agents. Soup isn't.