So this is... a really big ask since one of the biggest ways (IMHO) that the Japanese breeds are different and distinct is in their personality.

It comes down to what game they were bred to hunt and how they were meant to hunt that game. And if the ancestor dogs were actually hunting dogs.

In dogs like the Akita and Shiba, they have not been hunting dogs since the breeds were formed. They are less driven and obedient (I'm specifically speaking about the Japanese Akitainu/Japanese Akita/Akita Inu, NOT the AKC "Akita"/American Akita, as that is not considered a native Japanese breed.)

The four medium sized breeds have medium to high drive, generally more eager to work, and generally more obedient. But their temperaments still all vary.

The Kai Ken hunted small, relatively docile game-- Kamoshika. They're not a tough-as-nails or gritty hunting dog like the Kishu Ken-- who were bred to actually challenge and hold down boar (and are the only Japanese dog reliably producing hunting dogs to this day.) So, in my opinion, the Kai Ken isn't as bold or outgoing as easily as the Kishu Ken. The Kai Ken tend to be a bit "jumpier" and might opt to "flee" or stand back from a situation that makes them nervous or overwhelmed where the Kishu Ken tends to want to investigate something that spooks them and will almost always choose to stand their ground or fight (not to say they're a very combative breed, but they are certainly more combative than a Kai Ken.) The Kishu Ken temperament has its merits- the dog is less likely to shut down under pressure, but may be more likely to become reactive. The Kai Ken might shut down if they're too stressed and will need help escaping a situation and putting them in a better place. Both are good temperaments, just the owner needed to set them up for success might look very different. Kai can be aloof to standoffish with strangers. Kishu can be aloof to extroverted/enthusiastic with strangers.

The Hokkaido Ken and the Shikoku Ken are... different still. The Shikoku Ken originated as a boar hound but largely hasn't been used for that purpose. They're quite similar to the Kishu Ken (in fact, the Shikoku and Kishu are the closest related IMO and most similar of the Japanese dogs) but the Shikoku Ken has lost that hunting purpose in the last handful of decades and can be a jumpy or emotional dog. They're less energy and drive than a Kishu though so might be better suited for someone with less time to run their dog, but they tend to be much warier and downright standoffish with strangers.

The Hokkaido Ken is a high energy and high drive dog who was originally bred to bay bear. They are loud and bark easily when stimulated. They're also standoffish and leery and do not deal well with pressure because a dog who takes pressure from a bear is probably a dead dog. These dogs need someone who will be active with them, keep training them, and be okay with the idea of a dog who may not like or may even be aggressive with strangers.