These are all great questions. Let me tackle them one by one.
The origin of Pilates was a man named Joseph Pilates whose fitness ideas were rooted in yoga, Zen meditation and ancient Greek and Roman physical regimes. He was also very committed to what has become a popular New Age concept known as the power of positive thinking.
A big problem with Pilates is in the instructors who may or may not be teaching a version that incorporates New Age techniques. Examples are "Yogalates" and "Pilates with Chi."
Having been a fitness instructor for many years, I can say that Pilates is a resistance exercise of which there are many others. In fact, the gold standard (according to science) continues to be free weights (I use them 3x a week) because they build muscle which elevates the metabolism (good for weight loss) and build bone strength (good for post-menopausal women). Exercise tubing and bands are also excellent, as are the resistance machines found at your local gym. But free weights remain the best choice for overall fitness value.
As far as stretches are concerned, there are dozens of stretches used at the beginning and end of classes to warm up and cool down. These are not yoga moves. Normally an instructor will say if they are using yoga, and the only stretch I know of that even remotely resembles a yoga move is a back stretch where you lay on your back and pull the knees tightly to your chest. A similar move is called the "rock 'n roll" position in yoga and is used to transition from one pose to another.
You may want to review basic yoga moves (there are about 4 billion websites to choose from) just to familiarize yourself with what they look like.
Great questions, Margaret!