Seven-time world champion Lewis Hamilton and the FIA were in a standoff at this month's Miami Grand Prix over piercings the Mercedes driver has raced with for years and said he could not remove.
The sport's only Black driver, an outspoken champion for equal opportunities and diversity, Hamilton has also experienced racism and online abuse.
"I don't really fully understand the whole idea when people get married the woman loses her name," said the seven-time Formula One world champion.
Media reports have valued the new deal at 40-50 million euros ($44-$55 million) a year.
The Briton is being groomed to be their next world champion with Lewis Hamilton, 37, out of contract at the end of 2023.
Hamilton, the sport's most successful driver of all time, did not attend Thursday night's gala awards ceremony-- a requirement for the championship's top three.
Lewis Hamilton, beaten this year on the last lap of the last race, did not attend the event at the Louvre museum in Paris.
Red Bull's Verstappen seized the title after overtaking seven-time world champion Hamilton on the last lap of the race amid uproar over how a late safety car period was conducted.
Red Bull rival Max Verstappen ended the Briton's quest for an unprecedented eighth F1 title with a safety-car aided, last-lap pass for championship glory at the season-ending Abu Dhabi GP.
Verstappen passed title rival Lewis Hamilton on the last lap. After a record 22 races, Verstappen had 395.5 points to Hamilton's 387.5.