In iOS 13 Settings > Safari > Downloads there’s a setting Other… for where to save downloaded files. If I select that I see a menu of options, one of which is Dropbox. If I select it though the Done button gets grayed out. The only option I can actually choose (apart from various other apps) is Downloads on my iPhone which I don’t want. Has anybody managed to select Dropbox as the default save location?
No. Mine also has a OneDrive option that is grayed out. If I were to venture a guess, it’s that iOS 13 has opened this up as an option but it will require DropBox, Microsoft, Google, et. al., to make this option live. Or it is reading the availability of these locations from the Files app but that Apple has not yet made this an available option.
However, I can change the option to a location on iCloud Drive. (The default seemed already to be the Downloads folder on iCloud Drive.)