Are effect sizes and power the same? No. An effect size is extant. It’s real, it’s a thing that can be observed. But will you see it? That depends on the statistical power of your study.
Think of an astronomer searching for a distant galaxy. The galaxy is real whether she sees it or not, but her ability to see it will depend on the power of her telescope.
For a good introduction to effect sizes – how to measure them, report them, and interpret them – check out my book Effect Size Matters: