What's going on with these tag types?

See post #736419 and post #736284. Tags like kaguya_(shining_hearts) show up with the correct typing in tag searches (and sometimes in tag histories) but show up as general tags in the individual post pages, and tag edits don't correct this.

They all seem to be recently-created tags, but I didn't think cache lag extended to tag types...

It does. The tag type is cached, and different versions may be cached in different places. It'll fix itself after a while.

For laggy tag types I notice that if you alternate between, say, character:kaguya -> copyright:kaguya -> character:kaguya, it'll right itself right away.