In social situations, or in nature, people use the word "ruler" or "king" to refer to the strongest entity in that environment, even if the "ruler" doesn't make decisions that we'd think of a leader or manager.
King of the Jungle? A lion. Not an actual "king" unless it's The Lion King.
Kukui as the strongest trainer in Alola is something of an unofficial Champion, even if he doesn't have the title. So Ash defeating Kukui, who many perceive to be the strongest, would legitimize him as worthy of the title. It's very similar to how perception of legitimacy limits how much power the Crown practically wields in the UK, even though British monarchs have a lot more absurd powers on paper.
When Kukui announced the Pokemon League, he did so with the backing of all the powerful entities in Alola; the Kahunas and Aether Foundation. That allowed the tournament to be publically recognized as the formal successor to the ceremonial battle atop Mt. Lanakila, the culmination of the Island Pilgrimage. All of that was necessary to sway public perception.
I’m no expert on Japanese but Satoshi is consistently called チャンピオン, only been used as a title for Wataru, Mikuri, Adeku, etc. Due to the fact there is no established E4 and all Satoshi is the reigning Champion just like in the games.
Bulbapedia's article is misleading because it mixes up anime terminology with the game's.
Pokemon Blue:
It says 「ポケモン チャンピオン 」 right there.
In Orange, the winning title is apparently 「名誉トレーナー」 (
meiyo trainer)
Also, Drake's title is "Head Leader", indicating he's a Gym Leader, and Orange (while a real league) does not have a Champion.
So Ash is definitely the Alola Champion, but his grip on the title isn't that firm without defeating Kukui.
