core concepts and philosophy
Is reincarnation the same as transmigration of the soul in Hindu belief?
What actually moves between lives
Hindu philosophy draws a careful line between two things. The atman is the deepest self, pure awareness, unchanging and untouched by anything that happens in life or death. Upanishadic teaching holds that the atman never truly moves, changes, or is born and dies. It simply is. Then there is the jiva, sometimes called the embodied self or the living individual. The jiva carries personality, memory, desires, and the weight of karma. It is the jiva that passes from one body to the next, not the atman in any simple sense. So when the tradition talks about what moves between lives, it is pointing at something closer to the jiva than to an unchanging soul.
The image from the Gita
The Bhagavad Gita uses a plain image to explain this. Just as a person puts on new clothes and takes off old ones, the embodied self moves through bodies. The self doing the moving is real, but the bodies are like garments. This image is meant to show that what is deepest in a person is not destroyed at death, while also showing that the body is temporary. It is a teaching about continuity, not about a fixed thing being posted from one address to another.
Where the confusion comes from
The English word 'soul' carries a lot of baggage from other traditions, where the soul is often a fixed, personal thing that stays exactly itself forever. When Western writers first tried to describe Hindu belief, they reached for 'soul' and 'transmigration' as the closest words they had. The fit was never quite right. 'Reincarnation' and 'transmigration' both suggest the same fixed thing moving intact from body to body. Hindu philosophy, especially in its Upanishadic and Vedantic forms, says something more layered than that. It is worth noting that Buddhism, which grew in the same region, goes further and teaches that there is no fixed self at all, which is a different position from Hindu teaching.
How people talk about it today
In everyday conversation, most Hindus use 'reincarnation' and 'rebirth' freely and mean something real and important by them. The finer points about atman and jiva matter more in philosophical and devotional study than in daily speech. Different schools within the tradition also read these ideas differently, so there is no single agreed answer on every detail. What most share is the belief that something continues, that actions in this life shape the next, and that the goal is eventually to move beyond the cycle altogether.