Is supplicating for multiple deceased relatives in a general manner, such as "O Allah, forgive all our deceased and the deceased of the Muslims," sufficient when some names are forgotten, or is it necessary to particularize the supplication and mention each one by name?

Your supplication: "O Allah, forgive all of our deceased and the deceased Muslims" is sufficient and includes your deceased relatives. If you specifically supplicate for your closest relatives, then generalize to all your deceased and the deceased Muslims, this is an emulation of the supplication of the prophets, such as: {Our Lord, forgive me and my parents and the believers on the Day the reckoning is established} and {My Lord, forgive me and my parents and whoever enters my house a believer and the believing men and believing women}. And it is from the Sunnah of the prophets to begin by supplicating for oneself first, then for others.