I noticed Tiddlywiki has support to save to Github and Gitlab. But is there any way to save ti git running on my own server at home?
Git on its own does not have a REST API and therefore TiddlyWiki cannot save to a git server directly. You could look into packages that APIs to git servers and see if there is one that emulates the Github or Gitlab API.
If running your own server you can also consider WebDAV as a saving mechanism and set up a separate version control system.
AFAIK the git saver of TW also works with gitea, a frontend to git which can be self hosted.