The phrase covers two things that have very little to do with each other, and conflating them produces most of the confusion around it.
The stone structures
Stone circles with radiating spokes, built on the northern Plains of what is now the United States and Canada. There are well over a hundred. The Bighorn wheel in Wyoming, at nearly three thousand metres, has cairns aligned to the summer solstice sunrise and to the rising of particular stars. The Majorville cairn in Alberta has been in use for more than four thousand years, which makes it older than Stonehenge in its present form.
These are the property, in every sense that matters, of the nations whose ancestors built them and who use them now. Several remain active ceremonial sites, and access to some is restricted at particular times of year. They are archaeology only in the narrowest sense; they are in use.
The teaching wheel
A four-quadrant diagram used in contemporary Native education, counselling, addiction treatment and health programmes, associating the four directions with stages of life, aspects of the person, and other sets of four. It draws on Plains teachings and is used with full authority by Native educators, elders and healers within their own communities. It is a living pedagogical form, not an invented one.
It has also been detached from those communities and sold outside them as generic self-help, frequently with a colour scheme, an animal set and a list of correspondences assembled by whoever was selling it. The tell is uniformity: there is no single pan-Indian medicine wheel with fixed meanings, the associations differ between nations and between teachers within a nation, and any source that presents one canonical version as ancient and universal has built it.
The distinction that matters
A stone structure on the ground in Alberta and a printed diagram in a workbook are both called a medicine wheel in English. Only one of them is four thousand years old, and the age of the first is regularly used to sell the second.
Sources Indigenous Writes: A Guide to First Nations, Metis, and Inuit Issues in CanadaChelsea Vowel with The Medicine Wheel Revisited: Reflections on Indigenization in Counseling and EducationLloyd Hawkeye Robertson