No, you are confusing convention and simplifications for convenience for fundamental properties..
What is generally considered Standard set theory today is typically ZF(C), which can be viewed and was intentionally developed as what is not called a "material set theory" or sometimes a "membership based set theory"
In ZF(C), sets are characterized only by the binary “∈” operation and propositional equality of sets.
It is actually the less common schools like the Elementary Theory of the Category of Sets (ETCS), where individual elements have no identity and must be unique.
There are well documented disagreements between Zermelo and Frege vs Cantor and Lawvere, was the former was for membership and the later were against it.
With ∈ or "is a member of" both of the following hold:
Given a set A={1,2,3,4}; 3 ∈ A
Given a set A={1,1,2,2,3,3,4,4}; 3 ∈ A
There is value in defining things like multisets/bags and accessing concepts like multiplicity, but you could use a function you defined for a multiset and discard that information and it would fit the axioms of ZF.
In computer science and databases, this is less than just a trivial factoid depending how deep you get, it will directly relate to how you can trade space for time or reduce communication costs to save on time.
And yes even historically there were very popular network protocols where they would append the same element to a set and shift the complexity from insert time to read time.
To be clear I am not asserting any claim of better outside of local context to doing so.
You are absolutely correct that no production quality RDBMs are based on Codd's rules. That absolutely doesn't change the fact of what the term 'relational model' means, nor does it remove the reality that they follow enough of it so that where it breaks, even from a set centric perspective, is directly related to that origin. Especially in parallel, distributed systems that are the norm now.
And the fact that things moved, doesn't change the reality that the 'relation' in the 'relational model' is as described in previous posts.