From other answers that have been given, I understand that there is no prior review process, it has been requested more than once, but I don't remember seeing any official answer.
You also have to take into account that updating the game can lead you to have to be indexed again.
The process is not per account or per game, but apparently for each change, possibly to avoid malware or scams.
It is important to note that the page is public, that is, people can see/download/buy your game, even if it is not indexed, the only thing is that you will not appear in the searches within the itch website.
The best thing is that you create your own advertising, there is a post to promote your game here and it is always good to do it on pages or networks outside of itch and not depend to be indexed.