Stylus is an open-source fork of Stylish but without all the tracking. So yes, we use the same database.

* Access your data for all websites - required for the style elements to be added into the page, but STYLUS DOESN'T ACCESS ANY OF YOUR DATA.
* Access browser tabs - to be able to show styles in the popup
* Access browser activity during navigation - to be able to insert the styles at the moment the page is navigated

The permission names are generic, Stylus uses only a small subset of each permission. Please read some tutorial on how WebExtension permissions work for more details.