Yes, of course. Soil is just a medium which the plant can access nutrients and the micro-organisms that form part of the nitrogen cycle that it requires for its metabolism. It can be substituted for materials that can hold onto water and provide a habitat for the same micro-organisms.
If mangoes have been chilled in shipping, they will usually not make roots, like an avocado will if you suspend it over water with toothpicks. But I have had two mango plants grow just by sticking them into the soil of a large outdoor plant. Make sure the rounded end is pointed down, and plant the seed so that it is standing up in the soil. Same thing with water. Let the mango dry for a couple of days, then drop it into a glass of water, round end down, standing up, in a glass of water for however long it takes for it to split at the bottom and make a root, and split at the top and start a stem. Don't be disappointed if they don't grow, however. Best bet is to get yours from a farmer's market and make sure they haven't been refrigerated. Oh, and change the water periodically; it gets smelly.
No it has to have soil to live