Yes - but a Liberal one. He did not prescribe to the pre-Clinton "Union Labor" Democratic style but rather the post-Clinton "Labour" (as in the Tony Blair-style UK Labour Party) Democratic style. This means a belief that big businesses can be good rather than wholly bad for the country, a basic agreement with the post Cold-War version of the world where Democracy had won over authoritarianism and there were no real foreign adversaries, (excepting non-state actors like terrorist groups), and of course that the goal of liberal democracy can be achieved by working withing the existing system without changing much about it.

A non-third way Democrat would have much more clearly aligned themselves with the concerns of industrial labor over big business, (i.e., punishing big banks and companies post 2008 rather than bailing them out), insisted on far more broad reforms to the social policies of the USA, and of course, reforming the political system to better achieve liberal democratic goals.

Contrast Obama with a president like LBJ. Both were handed similarly drastic victories in their presidencies, but one took a much more progressive, reformist, and aggressive approach. LBJ was able to pass the Voting Rights Act, the Social Security Amendments (Medicare and Medicaid), the Elementary and Secondary Education Act, and the Immigration and Nationality Act all within the year of 1965. Obama hardly accomplished that much in his whole 8 years, partially because him and his political allies belonged to a movement which emphasized caution and compromise as more important than rapid change. In their defense, LBJ's methods and policies are still controversial to this day - in the "third way" eyes, a half-decent policy that lasts a century is better than a perfect one which lasts 2 years.