Why would a teacher say that meiosis is actually mitosis happening twice?

Because technically it does. Prophase-Telophase occurs twice, but Interphase occurs only once in meiosis. The cycle repeats in a loop, but it happens.

Because it is actually the process of cell division happening twice. the only phase of mitosis that doesn't occur twice in meiosis is interphase, when the chromosomes replicate. that only happens once so that cells can give half of their DNA to the gamete