I figured it out. The problem only occurs with pictures taken by my Lumia 950, specifically those that use Rich Capture. Photos taken in manual mode, or with my older Lumia 920 which doesn't support Rich Capture, do not have this problem and can be overwritten just fine. I only switched phones recently, which is why I hadn't noticed the problem before.
So basically the Windows 10 Photos app does not allow saving over images taken with a Lumia phone in Rich Capture mode:
The solution is very simple - rename the file to remove the "Rich" part from the filename. This causes the Save button to reappear:
The following PowerShell snippet can quickly rename all files in a folder:
gci | ren -newname { $_.name -replace "_Rich", "" }
C:\Temp