on its release day, that is an indexing issue
As I have said above: There is no such thing as a release date on Itch.
There is no greenlight phase, no early access, nothing of the sort. Your publishing date is your publishing date. The release date you can enter is cosmetic. Unless you have a paid game, then people cannot buy the game if the date is in the future. But it is still published.
but they only work for the search bar match, not for the category filters.
I have no idea what you are talking about. You did not add any more tags to your game. And games are not searchable via tags in the search bar.
https://itch.io/search?q=Dungeon only will show games that have Dungeon in the title or in the short description under the title, but it will not show games out of https://itch.io/games/tag-dungeon nor https://itch.io/games/tag-dungeon-crawler . Also, search results are capped. You will not see all games that have dungeon in the title, only a page or so.
I’ll just wait for the manual review to eventually clear.
There is nothing to clear. Your game is indexed. If it was previously not indexed, that was already cleared up.