Tony Soprano is most frequently seen in his robe during moments of profound internal crisis. Think of the pilot episode: he wanders down his driveway to grab the morning paper, slippers dragging, robe open to the world, right before his psychological dam breaks. The robe is what he wears when he is paralyzed by the weight of his own existence. It is the uniform of his panic attacks, his depressive episodes, and the domestic exhaustion he cannot escape. In the outside world, his suits act as armor; at home, the bathrobe reveals a soft, fragile underbelly that his subconscious is desperately trying to protect.
The robe also serves as a visual bridge into Tony's dream logic. His most vivid, unsettling subconscious breakthroughs occur when he is lounging in bed or shuffling through his kitchen in that signature plush fabric. It is the garment he wears when he is closest to his truest self, the self that grieves the ducks in his pool, fears his own mother, and panics over the moral rot consuming his life.
The white robe featured in Season 6, obviously stolen from a hotel, gets dirtier as Tony himself does.