So Familiar, Yet So Misunderstood

Question

Anna says:

You have those I ask for size
Though not everything inside
Completing the order of queue
That's how I acquire you

Alex says:

Number of queues I cannot count
Joined to pose you, a new fount
Another copy of you
That's completion to go through

Tommy says:

Already so complete own
No breakage when let alone
Ironic, 'cause of the queue
Totally broken in you

What are you?

Subtle hint:

Focus on how "queue" is pronounced.

Moderate hint:

For each speakers of the poem, the first two letters of their name are a clue.

Answer 1

In the context of this puzzle I am

THE SET OF REAL NUMBERS

Title:

A lot of us non-mathematicians are familiar with the term REAL NUMBERS but often have misconceptions as to what they are.

Hints:

1)

This is pointing to the significance of the sound of queue.

The first two letters (AN, AL, TO) of the speakers names are clues which hints at different kinds of Mathematicians. They are an Analyst, Algebraist and Topologist respectively.

And so…

Anna says:

You have those I ask for size

Mathematicians ask for the size of the subsets within it. With the Lebesgue measure, the set of real numbers becomes a measure space, where subsets of are assigned a non-negative real number representing their size.

Though not (for) everything inside

Certain subsets cannot be asked for size because their size is not measurable.

Completing the order of queue

That's how I acquire you

Completing the ordered field of Q (set of rational numbers) you obtain the Set of Real Numbers - a Complete ordered field. This can be achieved though Dedekind Cuts.

Alex says:

Number of queues I cannot count

Joined to pose you, a new fount

This means to identify R as a vector space over Q.

Another copy of you

That's completion to go through

This is talking about the fact that another copy of R is needed for algebraic completion, to acquire C – the complex numbers.

Tommy says:

Already so complete (on its) own

No breakage when let alone

This is referring to complete metric space.

Ironic, 'cause of the queue

Totally broken in you

This means because of the set Q having metrizable space disconnected within it, so space is also disconnected within R since Q is the subset of it.

Answer 2

I think it is possible the answer is two parts, one general, one specific:

I think the general answer is:

Protein

More specifically:

The protein called "Alpha-1-anti-trypsin" (AAT)

If you find yourself asking "What on earth is that?"

Here's a link that has an abstract on what it is and what it does.
In summary: AAT is the name given to a protein that scientists are finding has many functions including reducing inflammation in the lungs and binding the enzyme "trypsin" (an enzyme that is good when present in the small intestine, but not in the lungs.)

First it satisfies the title:

So Familiar,

We are well familiar with protein as we consume protein on a daily basis.

Yet So Misunderstood

Yet we misunderstand that protein is not simply a single nutrient, but there are a wide range proteins in our bodies that serve the purposes of breaking down, disposing of, or binding up enzymes. (Many do this by reading and splicing DNA.)

And that leads us to the riddle itself.

First you may note the characters speaking "Anna", "Alex", "Tommy",

add up to the initials "AAT" (the initialized name of our protein)

You have those I ask for size
Though not everything inside
Completing the order of queue
That's how I acquire you

I believe (as this is "Anna" speaking) it refers to the first part of the name of the protein "alpha-1", which is a term "[referring] to the protein's behavior on protein electrophoresis" (Source) which is a detection and separation method where proteins are separated and ordered by characteristics Completing the order of queue and aquire[d] in the blue strips.

Number of queues I cannot count

The blue strip method (as pictured above) of acquiring the "anti-trypsin" does not tell you how many proteins there are (hence I cannot count.) And "Alex" being the speaker, refers to the second initial of the protein name "anti" directing us to "anti-trypsin.".

-More detail on the above, if you're interested:

"Anti-trypsin" was the original name given to the protein until they discovered different characteristics of it that separated "ordered" on the electrophoresis (blue strip) test into "Alpha-1", "Alpha-2", and other quadrants of the test. The particular "anti-trypsin" that (by some characteristic) separated into the "Alpha-1" section of the test was dubbed "Alpha-1-anti-trypsin" (AAT). It is also sometimes called "A1AT"

Joined to pose you, a new fount
Another copy of you
That's completion to go through

Other methods must be joined to find a count (hence a new fount) and Another copy of you referring to the fact that another sample must be taken as the testing done before was destructive, and you have aquire another sample to do further testing (hence That's completion to go through) to finish the necessary testing.

Already so complete own
No breakage when let alone
Ironic, 'cause of the queue
Totally broken in you

I believe with the speaker here being "Tommy", that this refers to the last part of the protein name "Trypsin", which is the enzyme afterwhich "anti-trypsin" was dubbed because it binds and permanently disables Trypsin. Already so complete own - Trypsin is it's own enzyme, not technically part of the protein even though it's part of the protein name. No breakage when let alone - when AAT is not present it doesn't "Break" the enzyme. Ironic, 'cause of the queue and Totally broken in you - when testing for an enzyme, testing is usually destructive, the "queue" referring to the electrophoresis (blue strip) test mentioned above. And yes, it is ironic that when testing for both anti-trypsin and trypsin, one kills the other, but testing for them both destroys them both.

Part of me suspects there is another answer, if not just some detail missing, but this fit so well, I had to post it.