In international astronomy, there is one organization that has the authority to name astronomical objects: The International Astronomical Union (IAU). Names defined by the IAU are the primary names used by astronomers when doing formal research.
Official names come from a variety of languages, notably Arabic and Greek (these cultures were early to create written catalogs so their names have stuck).
Why do we need "official" astronomical names?
We use official names for practical reasons. Astronomers speaking different languages need an efficient way to share information. Having one official name per object is the only sensible option. Other names can be included in research and media but if you want people to be able to search and find your information, always include the official name.
Unofficial naming organizations
Some businesses and non-profit organizations offer the chance to name a star or other object. This is not official and in most cases it's a scam. Any organization is free to create their own database of astronomical names but they aren't likely to be recognized by anyone outside that organization (and their customers).