I miss a lot of information that are quite well exposed/explained in the issue report and would be worth reused in the docs: what a style database? Why would I want to use it? What types? Where are they stored? And this issue should necessarily concern also the [Style manager chapter](https://docs.qgis.org/3.34/en/docs/user_manual/style_library/style_manager.html) on the GUI, the save symbols widget options,...

I can't see such a button as part of the Layer styling panel. The "Save symbol" is an element of symbol properties (so available wherever you edit a symbol), mentioned at https://docs.qgis.org/3.34/en/docs/user_manual/style_library/symbol_selector.html#configuring-a-symbol. And adding the symbol to project style db is one option of the button, not the only one.