The developer tools in Google Chrome have an Application panel.
This tool contains the page's manifest etc. But I can't find this tool in Firefox. Is there such a panel in the Firefox DevTools?
For the manifest part, no, there isn't, because Firefox doesn't implement Web Manifest yet.
But the Firefox DevTools support the other parts of Chrome's Application panel, they are just placed somewhere else.
The storages (cache, cookies, Indexed DB, Local Storage, and Session Storage) can be inspected in the Storage panel.
Service Workers can be inspected on about:serviceworkers
.