Hcfyk

I Just want to know the name for this if there is one because I don't think it satisifies any of the formal definitions for sets, n-tuples, sequences, combinations, permutations, or any other enumerated objects I can think of.

For convenience, I will henceforth use the term $mathbf{ set^*}$ with an asterisk to refer to what I described in the title.

As a quick example, let $mathbf{A}$ and $mathbf{B }$ be $mathbf{ set^*}$'s where $$mathbf{A = {3,3,4,11,4,8}}$$
$$mathbf{B = {4,3,4,8,11,3}}$$

Then $mathbf{A }$ and $mathbf{ B }$ are equal.

If you're looking for something like a set which may have repeated elements, standard terms are multiset or bag. See multiset on wikipedia.

The common term is multiset. For a formal definition, you can for instance define the set of multisets of size $n$ of a given set $A$ as $A^n/mathfrak{S}_n$ where $mathfrak{S}_n$ acts by permutation of the factors; or if you don't want to be bothered by size you can define it as a map $f: Ato mathbb{N}$ where $f(a)$ is supposed to represent the number of times $a$ appears in the multiset.

These are two interesting models for different situations, and there are probably more.

In this context you can identify what you call a $mathbf{ set^*}$ with a function that has a finite domain and has $mathbb N={1,2,3cdots}$ as codomain.

$A$ and $B$ in your question can both be identified with function: $${langle3,2rangle,langle4,2rangle,langle8,1rangle,langle11,1rangle}$$Domain of the function in this case is the set ${3,4,8,11}$.

If two objects can be distinguished by the number of times an element appears in them, that is called "multiplicity". So the more mathy version of "finite collection of unordered objects that are not necessarily distinct" would be "unordered finite collection with multiplicity" or "finite collection with multiplicity but not order".

The single-word term for unordered collections with multiplicity is "multi-set", but I don't think there's any single-word term for finite multi-sets. Googling "math collection multiplicity no order" returns http://mathworld.wolfram.com/Set.html and https://en.wikipedia.org/wiki/Multiplicity_(mathematics) , both of which mention multisets.

Another term that is used in the context of eigenvalues is "spectrum": the multiplicity of the eigenvalues is important, but there is no canonical order (other than the normal order of the real numbers, but that doesn't apply if they are complex). When you diagonalize or take the Jordan canonical form of a matrix, it matters how many times each eigenvalue appears, but putting the eigenvalues in a different order results in the same matrix, up to similarity.