Can I create a fictional map?

OpenStreetMap the data project is a single global map and is for real world data, so no.

With quite a bit of work, you can use the OSM.org software to setup your own instance of the map. It's probably a better idea to explore something like QGIS a bit first. You can pull data from OSM, edit it and then save it locally. You could also use JOSM for that, but it has less support for exporting a map (vs data).

https://opengeofiction.net/ uses the OSM software to create fictional territories. Control of the map is somewhat curated, so you'd have to find out if your project fit in and so on.

Yes, it is possible.

I used to draw fictional towns with Merkaartor and JOSM. But don't send it to the server, of course. OSM editors are more straightforward at digitizing and the features you add are quickly rendered according to your map style. In Merkaartor (for JOSM it's more complicated) you can easily customize map styles if you're not satisfied with the existing ones. If you want even more, you can export your data and create the map with QGIS or other GIS software.

Have a wonderful week-end!

There are many methods for creating fictional maps using modern mapping tools. This is the wrong place to ask I think. Try a simple internet search on "fictional GIS" for a host of conversations and ideas. It might be possible to utilise OSM data as a basis for a map, or to utilise tools developed for working with OSM data, but your question isn't going to be about Openstreetmap itself. I suspect that most of those using modern tools to create fictional maps will be aware of OSM - it's these people you'll want to get help from, not the OSM contributor (etc) community. Good luck.