You aren't crazy. Every single interaction Yosuke and the protagonist IS that loaded. Yosuke Hanamura is in fact one of the most bonkers characters ever created; every single thing he says-- why would he say that??
The big gay elephant in the room: Yes, it is true that Yosuke did have a romance route that was cut from the game. Atlus have never adressed it ever, but the recorded confession dialog present in the game's files (from his social link and not intended for say, another character), the cut reversed link dialog for him and the other romanceable options, and the fact he still has specific flags to trigger a hug in his Rank 8 event are enough of a smoking gun. The cut dialog is from either his rank 9 or 10 (likely 10); ranks 1-8 were left as is as far as we can tell. There is a misconception that they took the romance out and substituted homophobia in (ie the camping trip, etc), but there is absolutely 0 evidence of that and it is far more likely he was intentionally written with internalized homophobia from conception.
Why do we call it souyo? The sou- comes from Souji Seta, the name given to the protagonist in the manga, and yo- comes from Yosuke. The Persona 4 community (particually on livejournal) had been referring to him as Souji for years before he ever got his canonical name of Yu Narukami; many fic writers stuck with Souji and it's why we still call it souyo today.
[sidebar: Interestingly, he is hardly ever called Souji in Japanese fandom; the example name given in the instuction manual- Kosuke Tsukimori- is far more common to see than Souji. Additionally, the protag's nickname is bancho, a term that refers to a leader of a group of school delinquents (aka the IT) and fits in with Izanagi's general aesthetic.]
The Japanese name for souyo is 主花, 花主, or 主花主 (depending on preference of dynamic). The 主 (shu-) comes from 主人公 (shujinkou; protagonist); 花 is the Hana- in Hanamura (花村). So shuhana or hanashu is the romanization of their Japanese ship name!
The Korean souyo fandom has grown tremendously recently, so their ship in korean is 주하나 (pronounced juhana, but is esentially shuhana) or 번숙 (banseok; ban- from bancho and -sook, short for -suke).
The main meat of souyo: The starting part of souyo is natch Persona 4/Persona 4 Golden gameplay, especially Yosuke's social link. Yosuke is also involved with most main story events and pops up in other social links (Marie, Naoki, even Shu), usually doing SOMETHING out of pocket while he's there. So it's difficult to point to a couple of things and say these are the souyo touchstones of P4; more than any other romance link, their relationship is baked into the game. A playthrough (or a watch through) of the whole game is recommended to really understand the depth of their bond.
Beyond that, the souyo agenda carries into the spinoff games (again, often more so than the romanceable options). Here's the important canon interactions that you should at least watch a playthrough of:
♥ P4 Arena: Yosuke's entire route
♥ P4DAN: the storyline; both souyo partner dances for Yosuke's songs + Signs of Love and Dance souyo partner dances for Yu's
♥ PQ: souyo wedding
♥ PQ2: the Junessic Land dungeon + Shape of Bonds side quest
PQ2 was the last time we've seen either Yu or Yosuke (outside of Yu's optional boss battle in P5 Royal). The P4G epilogue is the last chronological sighting of Yu and Yosuke, as both the PQ games take place inside P4's timeline. While many other P4 characters were referenced in easter eggs in P5, we have absolutely no idea what souyo are doing in adulthood.
Outside of games, the anime is a real mixed bag, but souyo lovers DO eat so it's worth it to watch the whole thing at least once. The dub and sub are equally good if you like both voice acting casts (and both are excellent!). For an abridged version I recommend these episodes: 1, 2, 12-15, (22 if you don't want to jump right into Dec. 3rd) 23-26. Episodes 12 and 23 have the BIG P4 anime souyo moments and are must watches.
The Golden anime is next to worthless for souyo lovers.
The three manga adaptations of the games (P4, Arena, and Ultimax) all have some souyo in there. Ultimax has hints, P4 is on par with the main game, and well Arena does this:
As for fan works, the souyo fandom is very much still thriving! Souyo is the biggest P4 ship on AO3 (and still one of the biggest ships for Persona at large); the activity level is generally at least one new fic/fic update every day or every other day. Since P4 came out in 2008, much of the fandom was centralized on livejournal in the early days; communities like mayonakatv and the badbadbathhouse (very nsfw) allow you to see fan works from the very first days of souyo shipping (sadly time has not been particularly kind to the souyo-centric lj community magiciansfool). Likewise the souyo fanart community is going strong. Fanart is more active on twitter than tumblr (especially for the large community of non-Western souyo fanartists), and Pixiv is another great source for art (主花 and 花主). Persona fans may seem a little batshit at surface level, but souyo fandom has created a fairly comfortable niche with tons of cool people! Some of them have been around for the full 15 (and counting) years, some jumped on when P4 was ported to modern consoles (like yours truly in 2021), but we're all completely delusional...
In conclusion, souyo WILL rewire your brain and possess you if you let them ♥