I also ran into this issue (Thunderbird 52 and SOGo 3.2.9), though for me a contact's email addresses beyond the second one are lost only when I edit the contact, not unconditionally during sync.

I originally wanted to write a patch for this. However, my conclusion was that this would require to basically rewrite the email address handling code in the conversion from Thunderbird cards to VCARD format and vice versa. I just got started with the SOGo Connector code and didn't want to come up with something that wouldn't meet the design expectations of the original developers. And end up not getting merged

For a start, all email addresses beyond the second would need to be stored in additional properties (e.g. extraEmail1, extraEmail2, ...) when a VCARD is loaded from the server. When a VCARD is written back to the server, these properties would need to be included as extra EMAIL entries. Of course, that would still lose the tag (work, home, PREF), but that's not that big of a deal.

The additional properties (extraEmail1 etc.) should also be editable from an extra tab in the contact edit window. They should be displayed like the first two email addresses in the preview at the bottom of the Address Book windows. They should also be treated like them when searching for or auto-completing email addresses.