I changed my services about six months ago. I use Mailbox.org as well. It syncs with Thunderbird and my mobile device. I use LineageOS for MicroG on my mobile device and the DAVx5 app for synchronization with the native apps. I use the OX Drive app to access my storage in Mailbox.org.
For Thunderbird, I'll try out the Cardbook extension. I currently use Lighting but I suspect that some portions of the calendar, like birthdays, do not come across. I'm not sure if that is a limitation of Mailbox or the synchronization services. Otherwise the rest of calendar, adresses, and encryption work fine.
I used Posteo for awhile too, which was good, but prefer Mailbox.org for a complete Google suite replacement (email, addressbook, calendar, docs, and storage). It‘s missing an alternative for Keep so I use Standard Notes. You also get more aliases with Mailbox and can use your own domain. All that persuaded me to move from Posteo to Mailbox.
DAVx5 for Android
Lighting / Cardbook for Thunderbird