The banana plant is technically a herb rather than a tree, it's leaves and stems will die down each year at the end of the growing season, but the root will keep dormant below ground and shoot will reappear in the spring when the weather warms up, with any luck!
It's often referred to as the "walking tree" as it will come up in a slightly different place each year. With protection from frost, it is possible to protect the roots with it's own dead leaves or as Sandy says with a horticultural fleece, outdoors in the UK. It's likely to become too big for the pot, if you want to keep it long term, so you may want to find a sunny spot in the open ground and care for it in the same way.
The coconut palms are sold in their thousands and although they look very robust and exotic, these sprouting nuts are really just a seedling and usually last for one season, as a house plant, then that's it!
It likes high humidity and can be kept looking happier for longer with regular misting with water.
Good Luck - Chip