In Jainism, the idea you’re asking about is close to the core teaching about jiva (the soul).

• Jiva is life. Every living being—humans, animals, insects, plants (in Jain thought some say plants are alive too)—has a jiva. The body is just a temporary garment the jiva wears.
• There is life behind life: the soul (jiva) is eternal. It takes many bodies and goes through repeated lives (rebirths) because of karma. Each life is like a new coat the jiva wears, but the jiva itself remains the same eternal consciousness.
• Karma and liberation: actions generate karma that binds the soul to the cycle of birth and death. Through right conduct (ahimsa, truth, non-stealing, celibacy, non-attachment), right knowledge, and right meditation, one gradually burns off karma and can achieve moksha—liberation from the cycle of life after life.
• Shared view in Digambara and Śvetāmbara: both traditions affirm the eternal, conscious jiva and its goal of liberation. The basic picture (eternal soul, karma, rebirth, liberation) is common, though there are some doctrinal and ritual differences between the two sects.

If you’d like, I can explain how this idea appears in specific Jain scriptures or stotras, or relate it to a simple meditation/practice that helps you reflect on the nature of the soul.