I was finally able to reproduce the issue, big thanks to @smeijer for all his effort providing all necessary information, sql dumps and file copies! 
What a nasty bug… The issue is caused by Nextcloud running the app’s AppDataRepairStep twice. This only happens if you upgrade and enable the app at the same time (i.e. updating the app while it’s not enabled).
This issue will be fixed with Pico CMS for Nextcloud v1.0.1 (will be released this weekend). In the meantime it’s very easy to fix this issue: Restore your previous version of cms_pico so that no update will be executed, enable your old version of the app and then update the app.