Can I travel to Germany with Italian MAE special ID card issued by government to certain categories of foreign nationals?

From the description, this sounds more like proof that you are employed by the Ministry of Foreign Affairs and not in itsself a travel document.

This card can used togeather with your passport to all Schengen Countries.

This is something you should really ask your employer and not here.

Carta d'identità M.A.E.
(Identity Card issued by the Ministry of Foreign Affairs)

Mod. 11 (beige) Funzionari delle Organizzazioni internazionali, Consoli Onorari, impiegati locali, personale di servizio assunto all'estero e venuto al seguito, familiari Corpo Diplomatico e Organizzazioni Internazionali titolari di passaporto ordinario

(Model 11 (beige) Officials of international organisations, honorary consuls, local employees, domestic staff recruited abroad who have followed their employer, families of members of the diplomatic corps and international organisations who hold an ordinary passport)

NB: Models 6 (orange) and 9 (green) for, respectively, staff of international organisations who have no immunity and foreign honorary consuls, are no longer issued and have been replaced by model 11. However, these documents remain valid until the expiry date stated on them.

On the back of the identify card it is stated that the card exempts the holder from the requirement to hold a residence permit and, together with a valid travel document, entitles the holder to enter the territory of any Schengen state.

Sources:

• ANNEX 20 SPECIMEN OF CARDS ISSUED BY FOREIGN MINISTRIES OF THE MEMBER STATES - Part 1
• ANNEX 20 SPECIMEN OF CARDS ISSUED BY FOREIGN MINISTRIES OF THE MEMBER STATES - Part 2