Yes, he likes them as ideas as per his own statements (which leaves open the question of whether his policies and actions reflect said stated beliefs). According to the Newsweek 2016 article "RUSSIA'S PUTIN: I'VE ALWAYS LIKED COMMUNIST AND SOCIALIST 'IDEAS'", quoting his remarks from a speech at a meeting with members of the All-Russia People’s Front in Stavropol, Russia, January 25 2017:
“You know that I, like millions of Soviet citizens, over 20 million, was a member of the Communist Party of the USSR and not only was I a member of the party but I worked for almost 20 years for an organization called the Committee for State Security,” Putin said, referring to the KGB.
“I was not, as you know, a party member by necessity,” he said. “I liked Communist and socialist ideas very much and I like them still.”
In his speech, Putin insisted he was never just a “functionary” when it came to party matters and said the Moral Code of the Builder of Communism—a set of rules to be followed by all party members—“resembles the Bible a lot."
Additionally, political-economically, he sounds like a typical economically left democratic socialist rather than Leninist:
We need business to understand its social responsibility, that the main task and objective for a business is not to generate extra income and to become rich and transfer the money abroad, but to look and evaluate what a businessman has done for the country, for the people, on whose account he or she has become so rich. (source)
We still have a great amount of work to do in social development, including resolving one of the biggest challenges we face in this area, namely, reducing the gap between high-income earners and people, citizens of our country, who are still living on very modest means indeed. But we cannot, of course, adopt the solution used 80 years ago and simply confiscate the riches of some to redistribute among others. We will use completely different means to resolve this problem, namely, we will ensure good economic growth (source).
Politically, he's on record criticizing Stalinism as well as Lenin; and stated:
People in Russia say that those who do not regret the collapse of the Soviet Union have no heart, and those that do regret it have no brain. We do not regret this, we simply state the fact and know that we need to look ahead, not backwards. We will not allow the past to drag us down and stop us from moving ahead. We understand where we should move. But we must act based on a clear understanding of what happened. (Interview with German television channel ARD and ZDF, May 2005, Wikipedia sites non-existent-now Kremlin.ru page)
and praising democracy, including in Russia:
The U.S. is a very democratic state. There's no doubt about that. And it originally developed as a democratic state. When the first settlers set their foot on the continent, life forced them to forge a relationship and maintain a dialogue with each other to survive. That's why America was conceived as a fundamental democracy. (source)
and
Russia has made its choice in favor of democracy. Fourteen years ago, independently, without any pressure from outside, it made that decision in the interests of itself and interests of its people — of its citizens. This is our final choice, and we have no way back. There can be no return to what we used to have before. And the guarantee for this is the choice of the Russian people, themselves. No, guarantees from outside cannot be provided. This is impossible. It would be impossible for Russia today. Any kind of turn towards totalitarianism for Russia would be impossible, due to the condition of the Russian society. (press conference with Bush, 2005)
