The Mailing Name field in em Client does not synchronise in Google Contacts. It persists in em Client for a while and then disappears.
Exactly the same happens with my CardDAV account. The data remains in the eM Client contact until the contact is synced with the sever. Thereafter only fields available for the online contact are retained in eM Client.
It is a hassle but as a workaround, the few contacts that I have with the extra data not supported by my online account are saved in Local Folders.