'Fun' fact: the Doge of Venice was a dictator, formally elected by the rich and powerful families but really passed-down in the standard ways of autocratic succession, that is inheritance and violent power-struggles.
Mr Musk may like the word 'doge' both because he may feel it his rightful position in society and because the Medici _were_, often, great patrons of the arts.
The word is derived from the Latin 'dux', from which we get the English 'duke' and the Italian 'Duce'.