Mainly because to be a champion that represents multiple nations you have to actually represent multiple nations. To draw upon examples you've made:
Poppy and Kennen are both representatives of yordle kind as a whole while also championing ideals of the nations they preside in now.
Udyr spent a lot of time in Ionia and essentially performed great heroics for their cause and their rights to live as individuals in the wake of the Noxian invasion. If he had just stayed in Ionia for a time he wouldn't really be counted among their numbers, because he was essentially there to train and then would've left.
Sona is Ionian born and was a citizen of Ionia for many years of her life, up until her late teens, making her a citizen of said nation until her adoption in to Demacian nobility, which changed her nationality but not her ethnicity. Sona being Ionian would be more or less a pertinent and well known fact about her, and she likely is seen as a representative of both nations she has lived in as a result.
Urgot is a Noxian man who embodies the extensive potential of Zaun made techmaturgy, so he represents each nation to the other in a way.
Cassiopeia showcases the kind of magic one gets from Shurima and is now heavily embroiled in the current affairs of that part of the world, all while being Noxian.
To draw on some other comparisons: Karthus isn't a Noxian/Shadow Isles champion because his interests lie with the Shadow Isles solely. His history as a Noxian is not important to the him that exists now. Riven's probably living in Ionia right now, but that doesn't mean she represents Ionian ideology or the nation as a whole. Lucian and Olaf were involved in the eleventh Harrowing but they didn't all of a sudden decide to save Bilgewater and become heroes; they were there for their own ends and left soon afterwards.
Essentially for champions to represent multiple nations they have to actually represent more than one nation in one way, and not just be slightly involved with one of two. There has to be a solid reason as to why they'd represent two nations instead of just the one they are currently most strongly affiliated with, and being a multi-national icon in that sense isn't easy, especially between parts of the world that don't interact like Ionia and Freljord. In fact that point actually directly hampers a champion's chances of existing as a multi national champion; a Demacia/Noxus champion is unlikely to exist because of the relationship between those nations. A Freljord/Shurima champion isn't likely to appear because geography, culture and social factors mean people of those two nations are very unlikely to meet in the world naturally.
Essentially the champion has to be meaningfully involved with both nations, and most champions (and most people in general really) are more about their local nation and problems on the home front as opposed to those of other nations. Exceptions exist of course but they'll be few and far between, even amongst champions.
Hopefully what I said had some modicum of meaning to somebody.
is linked to both Bandle City and Demacia.
is linked to both Noxus and Shurima.
is linked to both Bandle City and Ionia.
is linked to both Noxus and Zaun.
is linked to both Ionia and Freljord.
is linked to both Demacia and Ionia.
is both Bandle and Piltover
is both Shurima and Void
is literally everywhere
is now Noxus and Shadow Isles
is weird, but you could think he is Noxus and Piltover related, even though he is Ionian
Both Zaun and Piltover
Both Bandle City and Noxus
Both Noxus and Shadow Isles
Both Noxus and Shadow Isles
Both Shurima and Void
Both Shurima and Void
Both Serpent Isles and Bilgewater