From what I’ve seen, it really varies by geography, shop type, and even sometimes within a single chain.
For example, in the U.S., chain hotels make their names as verbose as possible to attract guests. An individual hotel’s name might mention not only the brand but also the brand’s parent company (“by Marriott”), the city, and any nearby attractions (“Airport/University Park”). On the other hand, chain supermarkets almost never go by anything other than the brand name. The individual store has no name, only a number that appears on a receipt or the cash register’s LED display. Some hotel and supermarket chains operate on a looser affiliation model, allowing individual locations to name themselves more uniquely. In all cases, an ordinary layperson would consider the POI to have a common name, but this may vary from common practice in other countries.
The Name Suggestion Index project, which helps ensure consistent brand and operator tagging, generally includes only the brand name in name=*, but it makes broad exceptions for some countries, POI categories, and individual brands.
Some related recent discussion: