The short answer to your question is "I don't know".

The longer answer is that I have wondered this often myself, as someone who is a stereotypical STEM graduate who wanted to understand the humanities more. Related to not knowing what makes a poem a poem is the insecurity I feel about not understanding what makes a good poem.

On my quest to understand this, I learned a lot about metre in poetry and the real interesting thing there, imo, is that all poetry inevitably builds on, or breaks from the traditions of its time, and that what "counts" as a poem changes. I also found I enjoyed reading poetry more as I became more familiar with poetic metre — not just being able to define terms like "dactylic hexameter", but being able to feel it too. For Better For Verse was a fun tool to help practice that. As you emphasise, structure in poetry is more than just rhymes.

However, not all poems that utilise rhythm have metre. Even understanding metre and other structured poetry conventions can only get us so far in understanding free verse. I don't recall where I read it (I'll try to find it later), but I read somewhere the argument that free-verse is just as laden with conventions and tradition as any other style of poetry, but it's comparatively new so our understanding of those conventions is still emerging. That's pretty neat because it makes me feel like I'm on an exciting new frontier, which makes me feel less bad for not understanding what even "counts" as a poem.

The most important thing I have learned though is that although poetry can feel overwhelming and difficult to break into if you don't grok all of this stuff, you have as much right as anyone else to engage with poetry and enjoy it however you like. For every great poem that's conventionally structured, there are countless more that were written in defiance of, or ignorance of poetic traditions. Maybe this is something that you don't struggle with, but certainly I found that I was inadvertently gatekeeping myself; I've enjoyed learning all this stuff, but it wasn't necessary in order for me to realise that I don't need to be able to approach poetry like I'm in English class.

For example, you ask why this would be structured as a poem rather than a short story — there's a sense in which only the poet could answer that question. Otherwise, I reckon the general answer is that framing something as a poem affects how it gets displayed, where it gets displayed, and consequently, who reads it (and what frame of mind). This is a poem because presumably the author wanted it to be read as a poem would, and framing the same information as a short story would lose something of what the author wished to communicate. Looking at it at that angle, it's as if your own question is bounced back at you: how did you know this was a poem, and how did knowing that affect how you engaged with it? If you imagine this as a short story, how would your response to it be different? We have to use our imagination to convert it, because even getting rid of the line breaks wouldn't be enough to convert it to coherent, continuous prose, but perhaps we can pretend for long enough to better feel the impact of the poet's choices.

These questions aren't rhetorical if you don't want them to be — I'd love to hear your thoughts if you have any, because I also can't pin down what "counts" as a poem. I recall the Supreme Court judgement where the justice said they couldn't define what counted as "hardcore pornography" but "I know it when I see it". Poetry is similar, I feel. That doesn't mean we can't attempt to articulate it though, especially because as I alluded to, we don't have to write poetry to be a part of shaping the continuously evolving traditions and understanding of poetry.