Ownership of companies via shares is only for LLCs (which is a possibility but in practice uncommon) and Corporations and both Sole Proprietorships and Partnerships do not have shares.

That said, it is indeed possible for a company to have ownership by shares and not be publicly traded (i.e. not in the stock market) so for example a corporation with no traded shares would be a private corporation.~

This is what I found out for the US by searching for "types of companies" and "LLC shares".

The countries I lived in and owned companies in Europe have similar things but with different names (for example what a Partnership in the US is roughly what's called a "Society by quotas" in Portugal and is actually the most common kind of company).

In my experience people that never had their own registered business don't usually know any of this because they have no need to look into it.