This is bug in Android, and it is not DAVdroid specific.
You have to install DAVdroid into the internal memory. DAVdroid should not allow installation anywhere else (if it does, it is a bug).
It also happens when you reboot into “rescue” mode, where all user-installed apps are ignored. You loose all account data instantly. If you have your device rooted, put DAVdroid APK into
/system/app. Android will handle newer versions correctly (just like any other upgrade), but it will prevent accounts from disappearing (you will only use older version for a while).
Note that this means user will loose all data created since last sync. It happened to me when we were traveling around the Corsica (with Cal/CardDAV-Sync, not DAVdroid, but it does not matter) and I lost calendar entries created in last 3 days, including some bus schedules and planned routes for next week. It also meant I could not access my addressbook for next two days until I found some cafe with wifi hotspot and synced old data back. And what was the trigger? We were in hurry to catch a bus in the morning, so I let my phone to boot up in my pocket and menu key got pressed in bad moment, which made my phone to boot into rescue mode… Awsome experience.