After several transfer windows in which Los Blancos were linked with a move for Hazard, the 28-year-old finally made the switch from Chelsea last week, signing a five-year deal with the 13-time European champions.
Madrid has overhauled its squad after a disappointing 2018-2019 and Hazard is the biggest name to arrive at the Santiago Bernabeu so far.
The club has a history of acquiring the best players in the world having brought in Luis Figo, current boss Zinedine Zidane, Ronaldo and David Beckham during the Galactico era of the early 2000s, yet Hazard refrained from putting himself in that company at his unveiling on Friday (AEST).
"I'm not a Galactico, maybe one day I'd like to be considered a Galactico," he said.
"I'm Eden Hazard and I have to start afresh here, it's a new chapter. I think I'm a very good player."
In five of the last six years, the Ballon d'Or winner has been playing for Real Madrid, with Luka Modric succeeding Cristiano Ronaldo - now of Juventus - in winning the prize in 2018.
Hazard finished eighth in the voting last year and many believe he can strengthen his credentials for the game's top individual prize having swapped Chelsea for Madrid.
"Well the [current] Ballon d'Or winner plays for Real Madrid... do I think it's easier? I don't know," he said.
"It depends how we play. It's true they've been winning the awards for many years now but it's not a goal of mine to win the Ballon d'Or.
"I'm here to help my team, my team-mates and maybe one day I'll win it but it's at the back of my mind. It's more important the club win titles."
Hazard spent seven seasons at Chelsea but insists he is still only entering his prime.
"I don't think I'm joining the club late on in my career - I'm just 28... between 27 and 32, those are the best years for a footballer," he said.
"I know there's been a lot of rumours and I've been linked to this club for many years but I've enjoyed spells at Lille and Chelsea and now I'm joining the biggest club in the world and will hopefully continue to develop and progress as a footballer."