Compare when Picard said people have moved beyond the need for personal possessions to Kirk's collection of antiques in his apartment.
I thought the line was "the acquisition of wealth is no longer a driving force in our lives..."
Or something to that effect.
I mean, everyone on the ship had possessions. Picard had his tapestry, flute, and photo album. Even in the Roddenberry era of TNG Picard liked his "stuff", going back to Stargazer to get it.
In "The Neutral Zone" he said, "People are no longer obsessed with the accumulation of things. We've eliminated hunger, want, the need for possessions. We've grown out of our infancy." Later, he went on to say, "This is the 24th century. Material needs no longer exist."
Of course, you're right, too. Picard had things. He wanted his stuff. People undoubtedly still have things of practical and sentimental value in the 24th century (just try to walk off with Riker's trombone and see his reaction).
Maybe Picard meant people didn't accumulate wealth or things just for the sake of it
(or the power they think they may get from it). Power is no longer equated with wealth. There is no need for it or incentive to achieve it.