The parsing of the sentence in the question is ambiguous. There is no single, clear-cut way of looking at it.
Before directly addressing the actual sentence, consider this:
1. They are cooking apples.
- They → pronoun, referring to people
- are → auxiliary verb
- cooking → verb
- apples → noun
2. They are cooking apples.
- They → pronoun, referring to apples
- are → verb
- cooking → adjective
- apples → noun
So, which is it? What grammatical function does cooking serve? The answer is we don't know. It would be resolved by context, but the grammar of the sentence itself, as a single sentence, changes depending on how it is viewed or interpreted. Or how it's parsed.
The same holds true of the particular sentence in the question. What does it mean?
There is more than one way of interpreting it.
1. The more you have, the more you want.
→ If you have more, dependent clause then you want more. independent clause
But, as a comment under the question asked, how can the more you want be an independent clause, something that stands on its own? It can be if it's parsed to be:
The more you want.
→ You want the more.
→ You want more.
This simply changes the word order, and then drops the article that would not normally be used.
While the specific wording of the original and the restated version with the article are not idiomatic as sentences, they nonetheless would be phrased as the final version you want more if that's what the original was interpreted to mean.
Many sentences that appear unusual, or even wrong, are still acceptable in archaic forms of English, or where the traditional SVO (Subject Verb Object) order has been changed. You might not normally use them, but they don't actually violate syntax. The reversed word order of Yoda in the Star Wars movies is famous for rearranging words in an order that sounds strange and would not normally be used, but which is still technically grammatical.
Grammatical but strange such sentences are.
The problem with considering only the second half of the sentence in question, and considering it as an independent clause, is that the first half of the sentence is required in order to understand why it might be considered an independent clause.
If the only thing you are exposed to is the second half of the sentence, it's highly unlikely it would make sense to treat it as an independent clause; it's more likely it would be considered a noun phrase—or sentence fragment.
But when analyzing the sentence as a whole, and parsing possible meanings, you have to consider the entire context.
2. The more you have, the more you want.
→ Having more means wanting more. a single independent clause
It's easier to understand why this rephrased version is a single independent clause—because, unlike in the other interpretation, the wording here sounds natural.
My personal opinion (and it's entirely subjective) is that this is the least complicated way of parsing the sentence. Perhaps because of this, it's also the way in which I would normally look at it.
As in the analysis of the first interpretation, and how a changed word order didn't necessarily result in something ungrammatical, the fact that there's a comma in its original form doesn't necessarily mean that there's a dependent clause. It depends on how it's interpreted.
There is a third interpretation of the sentence that could be given, and, following from the second interpretation, it supports the idea of a comma splice:
3. The more you have, the more you want.
→ You have more, independent clause you want more. independent clause
But being a comma splice wouldn't necessarily mean it's an error. Some comma splices are stylistically acceptable:
I came, I saw, I conquered.
It's not a bird, it's a plane.
Comma splices that involve very short clauses, generally with the same subject, are often considered acceptable as a form of deliberate style. However, it's true that not everybody accepts them in any form.
Depending on if you accept any form of comma splice, and if you parse the sentence in the question in this way, then you might decide it's incorrect.
There could be other interpretations I haven't thought of.
Changing how you interpret (or parse) the sentence changes the grammatical functions of its components. Just as different interpretations of cooking apples changes the grammatical function of cooking.
Normally, sentences aren't as grammatically ambiguous as this.
There is no way of giving an answer that is unarguably the correct interpretation and, therefore, the correct analysis in terms of which parts of this sentence break down into what types of clauses.