Can app with Storage permission access all data?
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 /data/data/&lt;package_name&gt; and /sdcard/Android/data/&lt;package_name&gt;).