There is a distinction between a star name and a star designation.
Star names are a sub category of star designations.
Regulus is a star name, but the other identites of Regulus are more like star designations.
A star is usually referred to by its designation in some star catalog. Thus the star will be identified by giving its catalog name or abbreviation, and then by its number in the catalog.
Wikipedia's list of nearest stars and brown dwarfs shows the vast variety of star catalog designations.
The nearest star system is Alpha Centauri (Bayer designation).
Then Bernard's Star, also identified as (BD+04°3561a) from the Bonner Durchmusterung of 1859-1862.
Then Luhmann 16, a brown dwarf system discovered by Kevin Luhman.
Then WISE 0855-0714, another brown dwarf.
Then Wolf 359, number 359 in the Wolf catalog, also known as CN Leonis,it's variable star designation.
Then Lalande 21185, also known as BD+36°2147.
Then Sirius, Luyten 726-8, Ross 154, Ross 248, Epsilon Eridani, Lacaille 9362, Ross 128, EZ Aquarii, and so on and so on. There is Struve 3928, Groombridge 34, GJ 1961, Kruger 60, etc. as examples of various catalog designations.
There is a catalog of all know stars within a distance of 25 parsecs or 81.54 light years of the Sun, the Gliese Catalog of Nearby Stars, so all of the above stars also have Gliese catalog designations in addition to the the ones given in the Wikipedia list and in still other star catalogs.
The brighter a star appears as seen from Earth, the more names it will have in various Earth languages and cultures.
Star Names: Their lore and Meaning Richard Hinckley Allen, 1899, gives the names of many stars in various cultures.
Chinese star and constellation names are found in Stars of Jade: Astronomy & Star Lore of Ancient China.
This article lists some of the more important star catalogs.