Is it possible to publish unpublished software made by me on itch.io, even if not necessarily in the field of games, pixel art and tools?
Publishing unpublished software is the whole point of itch. You can even publish published software. Itch is not exclusive. You can sell your stuff on other platforms as well and even link to there from your project page. You will find many games with a link to their steam page.
Also publishing unfinished software. There is status prototype and in development to indicate, it is not finished.
Accounting software is tools. Basically every software that is not a game is a tool. That is the whole point of sofware, doing tasks. So even games could be considered tools, but the purpose is recreation.
The other stuff is not of legal nature. It is of terms of service of itch and community rules. Self advertising is not allowed in most places. There is that announcement thread and you can have devlogs and such. But there are also rules, you should read before posting.
But you should not expect much traffic coming from within itch to your accounting project. As you have observed, the people here are interested in all things around games. You have to advertise your stuff outside of itch.