TL;DR- Yordles are spirits helpful to mankind. Illaoi seeing one is an insult to her strength and fortitude. Tahm Kench wants to exploit mankind, so he rivals helpful yordles. Gregor from the Howling Abyss is insulted by a visiting yordle because their appearance before him would imply that he is not a proud Iceborn warrior, but a regular human. Yordles physically manifest according to their spiritual nature (militaristic ones will appear as soldiers, scientists will be exaggerated as such, and a righteous one such as Poppy will look like a Demacian soldier), which explains why they all look different.
In lieu of Tahm considering certain yordles to be rivals, and Illaoi seeing yordles as spirits...
I think it's plausible so say that yordles, in general, are viewed as helpful spirits by peoples outside of Bandle City. (Poppy is a good instance of this. She is trying to deliver a legendary hammer to a worthy hero, and would probably be viewed as a helpful wandering spirit trying to give away a fantastic weapon)
Whereas Tahm Kench (and his considered ally, Aatrox), seem to be more malevolent spirits that represent different thematics (such as Desire and Wrath)
Of Tahm's rivals, Corki, Heimerdinger, Rumble, Tristana, and Ziggs, all have some form of technology about them, whether they are pioneering it or using it. Tahm Kench hangs around Bilgewater now, and Bilgewater seems to be a bit more primitive in terms of technology, and there is a kind of mythological/spiritual culture about the place. Tahm Kench, being a spirit himself, would thrive here. The technological advancement of the 5 technological yordles contradict a culture of spirituality. Tahm Kench cannot thrive in a world where technology pushes out spirituality.
Lulu and Gnar were suspended in time, so they are technically very old. Perhaps Tahm dislikes them because they know of the world long ago, and may have an inkling of Tahm's true nature, before he assumed the form of the charismatic catfish.
Kennen, being a part of the Kinkou Order (I think that's what it's called??), is a fighter for balance in the world. Tahm thrives off imbalance in the hearts of people, so Kennen and Tahm's ideals conflict in this way.
Poppy, being attracted to Demacia and looking up to the likes of Garen, probably has some ideals pertaining to lawful righteousness and mental fortitude. Again, contradicting Tahm's ideals and success.
I don't know what to make of Tahm's relation to Teemo and Fizz.
And the bottom line of all of this: Illaoi recognizes yordles to be spirits who aid humans.
Another thing of interest is that Gregor (ghost shopkeeper on Howling Abyss), says to visiting yordles "A yordle?! Gah!"
What comes to my mind is that Gregor takes pride in being an Iceborn ("I am Iceborn!"), and having a yordle appear to him is insulting his being. Gregor says that the Iceborn "were human once", and he takes pride in being an Iceborn. Having a yordle appear before him would be to imply "you are not Iceborn, you are a regular human". So to Gregor, seeing a yordle is a bit insulting to who he is.
Setting aside the ideals of the above yordles and how they conflict with Tahm, reconsider their spiritual nature: helpful spirits to mankind. Tahm seeks to exploit men for the sake of his own gain, and helpful spirits would undo that. Yordles being spirits would also help to explain why they look so different from one another. They physically manifest according to their thematic/ideal, which may have some implications for Teemo and Fizz's purpose, based on how they look.
All just hypothesis though.
at various times. This is because of their races specific nature that makes them more perceptible to inner demons when socially outcast. When not, they are hard to be manipulated, which is why almost all Yordles are seen by this demon as rivals."
