Downloaded files w/o Files app

if I didn’t have the Files app?

You do have Files app, use App Library to search Files app, see this Apple article:

Organize the Home Screen and App Library on your iPhone

Organize the Home Screen and App Library on your iPhone - Apple Support

If you cannot find Files app, download from the App Store, see ->

https://apps.apple.com/us/app/files/id1232058109

It’s a document…pdf file. I downloaded the file from my phone but cannot find it anywhere.

Use iCloud Drive, see this Apple articles below:

Set up iCloud Drive

Set up iCloud Drive - Apple Support

Access and manage your files in iCloud Drive

Access and manage your files in iCloud Drive - Apple Support

What type of Data are you trying to transfer? Where do want to store it? Do you have a compatible App for storing the data/image/Music being AirDropped?

For example, if you are AirDropping MP3 files.

The storing process entirely depends on the apps that you have on your device to store and play mp3 files.

See the pic below. I have many other apps which can store and play these files including Apple Files App. If you only have one app say, WhatsApp then there is nothing for the iPhone to get permission for, from the user.

It will open WhatsApp and Store it there else it would seek your permission as in the GIF below.