When I give Storage Permission to an app, is it true that it is able to access all data on the internal and external media, for instance the Downloads folder? Or does it only get access to store and retrieve its own data in a sandboxed environment?
Yes. The storage permission grants an app access to the whole "Internal SD Card", where all your photos, musics, downloads reside.
It's usually safe to assume that apps you trust access only what they need, but a random unknown app may scan your files and expose some privacy problem.
An app doesn't need any special permission to access its own, sandboxed data (usually