The Iliad and the Odyssey are considered poems because they are written in meter (a set number of syllables per line with a set pattern of stressed and unstressed syllables in lines). Specifically, the poems are written in Dactylic Hexameter. Meter is usually broken into feet (a set subunit of a line). Hexameter means that each line has six feet. Dactylic explains the pattern of stressed and unstressed syllables. The way a dactyl works is that it's one stressed followed by two unstressed syllables. So in theory it would go:

BA-dada BA-dada BA-dada BA-dada BA-dada BA-dada

That said, dactylic hexameter was not quite so strict and the first four feet could be replaced by a spondee (two stressed syllables). In addition, the last foot was almost never a dactyl and instead would always be a spondee or a trochee (one stressed and one unstressed). This meter was the most common for epic poetry and can be seen in the Iliad, Odyssey, and Aeneid (which is Latin and much later). Other genres of poetry had different meters, sometimes the same meter. The idea was that certain meters helped express certain topics better. This is something we still use: one wouldn't want to use the rhythm and melody of "Twinkle Twinkle Little Star" to write a song about death (unless you wanted to be ironic or comedic).

Not all classical literature was poetry. Philosophy, history, oratory, novels, architecture manuals, and so on were all written in prose (i.e. without meter). Prose sometimes uses metrical conventions for rhetorical purposes, but the entire work is not governed by meter the way poetry is. So, the Iliad is a very long text that is all written according to the strictures of meter. Herodotus's The Histories or Plato's dialogues are not written in meter. The problem is that in translation, it's hard to copy meter and so a work like the Iliad ends up looking a lot like The Histories. Even keeping line breaks in the same place is very very difficult because sentence structure is different between different languages. And even if one wanted to keep the meter the same, it would be weird to read an English poem in dactylic hexameter because it's very uncommon for English poetry. English poetry tends towards iambic meters. There are examples where the translator will adapt the work to a native meter or rhyme scheme (famously Alexander Pope translated the Iliad into heroic couplets), but this usually requires a reworking of the text, and so makes the translation less faithful to the original.

As to 19th century poetry, well, I guess it depends on where and when? It's not at all a monolith (poetry from 1810's England isn't the same as 1850's France or 1890's America). By and large, English poetry in the 19th century still stuck to some form of meter and rhyme scheme, but it was beginning to break down in favor of free verse. Now, rhyme is something that I've just brought up but didn't mention with regards to classical poetry. In Latin and Greek, rhyme was not necessary for poetry. It could be added as a flourish, but it wasn't constant. Traditional poetry in the Romance languages tended to view rhyme as necessary. In English poetry, rhyme was important but could be disposed of depending on the genre and purpose of the writer (Milton did not use rhyme in Paradise Lost, for example). Basically, every language has its own conventions of what is important in poetry, and these conventions evolve over time. But I do want to stress how important language is here: different languages fit different meters and the concept of rhyme better or worse than others. English and German are harder to rhyme than French or Italian, and so rhyme is less stressed and less absolute in the first two than last two. Latin and Greek fit dactylic hexameter well, English fits iambic pentameter well based on how each language pronounces words and sounds. That said, if one were showing an English poem from Shakespeare (late 16th/early 17th century), Pope (early 18th century), or John Keats (early 19th century) to an ancient Greek (with a time machine), they would recognize the meter (they would probably wonder why the author chose iambic pentameter so much) and would be confused by why they felt the need to rhyme so much. They would probably recognize it as poetry because of the meter.

In the 19th century, though, rhyme scheme broke down, as did meter. Poets felt constrained by it and wanted to write in a more natural or unnatural way (it depended on the poet). Some wanted to break down all convention, have awkward line breaks and spacing of words, some wanted to write in a way that was naturalistic. Through the twentieth century, at least in the United States, you see a gradual move to free verse to the point where the vast majority of poetry is written today without any consideration of meter or rhyme. Whether or not free verse counts as poetry depends on how conservative/absolutist one wants to be. There are nearly no scholars of literature going around and saying that poetry no longer exists because it's almost all in free verse (there are probably none, but I don't want to be too absolutist here). That said, if you showed a poem in free verse to an ancient Greek (again via time machine), they would probably say that it isn't poetry because it has no meter. Modern poetry is more about figurative and symbolic use of language than it is about structure. To ancient Greeks, medieval Europeans, and early modern Europeans, poetry was about form. Forms could be and were broken, but usually only sparingly and for specific purposes. The differentiation between prose and poetry was form not content. That is not so much the case today, where "prose poetry" exists.

As to modern music, I guess it's arguably more poetic from a form perspective than free verse poetry, but it often doesn't worry about meter that much. I'll say, though, that this analysis is hardly rigorous on my part. Music tries to fit lyrics to rhythms and melodies and this is very similar to meter, but in a song, syllables can be held over multiple notes allowing for it to fit the rhythm without having the same number of syllables in the line before or after. If you read lyrics and try to fit the rhythm, you'll often find yourself having to stretch or compress syllables to fit it. When reading without music and without pitch, this feels pretty weird. When singing, though, it's natural. Rhyme scheme is often present in music, so that does make it fit a bit more to the form of traditional poetry in English, but to an ancient Greek, they wouldn't really care about that. They'd look at the meter and see whether it fits a form, and I'd guess that most music doesn't fit into meter.