Buddhism provides a description of the soul as fleeting, ever-changing, and impermanent. It does support the idea that there is an essence that makes one who they are, and that this essence is separate from the body. The soul, to the Buddha, is a continuous stream of mental states. Our minds and experiences exist as a flow of our thoughts and conscious states, and this flow is both never-ending and continually transforming. No thought is ever the same as the last nor the next, and no conscious state can ever exist again after the brief flash in which it is conceived. The essence of our being is only "us" in that we are a continuous entity, but that entity does not have any permanent or foundational qualities. The concept of anatta, "no-soul," refers to the Buddhist idea that we do not have a permanent soul because we are constantly changing. If we conceptualize the idea of the soul as an eternal definitive piece of a person, that does not exist in Buddhist teachings because no part of life is eternal. This comes directly from the idea of life as a series of fleeting mental states; life is defined as ever-changing, so no one thing can last forever (or even longer than a flash). However we do have an everlasting essence, but that essence never consists of the same thing as any moment before. Overall despite the belief in a form of soul, the Buddha discourages focusing or even thinking about the soul. Trying to define one's essence, understand it, or speculate upon it just leads to confusion and rambling. The Buddha argues that we should not seek to understand or improve ourselves, but instead focus on what is causing us to be unhappy with that self in the first place. Focus on the soul is a fruitless use of time; focusing on our suffering and its causes and eliminating that suffering is a better path to happiness.
The point of a soul is that it defines a person and makes up who they are. If, as Buddhists say, the essence of a person is constantly changing, it doesn't make sense to define this as a soul. A soul should be a fundamental, unchanging part of someone, so not being constant disqualifies the Buddhist concept as a soul.
[P1] Our essence is an everchanging flow of thoughts and perceptions. [P2] We do not have an eternal soul, but instead exist as an impermanent but never-ending continuum.