A beet is technically a root. So the root of the plant is the beet, and whatever that grows out of it is the actual 'plant' portion.
Tap root
root
root
It is a Tuber
a root
They are the same, in most cases. Red beet is the common name for beet root sold as food. Beet root is more generic, and can refer to other members of the beetroot family, such as sugar beets.
potato head.
beet root
Turnip is a root vegetable that looks like beet. Beetroot Sugar Beet Parsnip Swede
The answer will depend on where in the world you are trying to buy it. Since the location is not identified, it is not possible to give a sensible answer.
A fruit contains seeds. A 'beet' is the root of the beet plant, and is therefore a vegetable, not a fruit.