Vectors in this case are more like an array of numbers, often in the thousands, where each number represent a concept associated with the word. So for example, "elephant" will have have values in the cells representing "animal", "grey", "big", "noun", etc... but low values in "verb", "abstract", "flight", etc... The meaning of these cells is usually not explicit, it is something that emerges from the model learning process, but since these models are usually trained on human language, you often find human concepts in there.

And indeed, mathematically, they are the same as vector in physics, just with a lot more dimensions. And indeed there is a concept of magnitude (norm) and direction. Some processes only use the "direction", throwing away the magnitude through a process called normalization, some don't care, and some actually make use of the magnitude.