Can’t save file “to my phone?

Yes, the reason that there are no options under the "On My iPhone" is that, natively, iOS does not have a file management system. The Apple Files app will only save to this location if the apps creating the files are either Pages, Numbers, or Keynote. Since a PDF is not one of these, there is no option to save it here. Instead, the Files app relies on cloud services to save files to.

Again, this is why I use FileBrowser. You may want to consider doing the same.

Ok, I just verified the following worked for me. I have an iPhone SE running iOS 12.4.

• Tap on the attached PDF file in the Mail app mail message. This should "open" the PDF for viewing.
• Tap on the small box with an upward pointing arrow in the lower-left corner.
• Tap on "Save to Files."
• Tap on the location that you want to save it to. I chose "On My iPhone." I my case, I had two locations to save it to: FileBrowser and HP Smart. (Both of these apps are on my phone.) I then tapped on Add.
• When I open the FileBrowser app on my phone, I can find the PDF in the "On My iPhone" folder.

Where you can save a PDF file to will depend on what you have available on your iPhone. In my case, I rely on the FileBrowser app. The native Apple Files app will only allow you to save iWork files directly to the phone.

Ok, since you set the "ground rules." What app (Keynote, Numbers, or Pages or something else) created this file, where is this file located (on a device on the local network or from the Internet) and what app (the Apple Files app, FileBrowser, or something else) are you using to try to save it to your iPhone?