This character is a Other Punctuation and is commonly used, that is, in no specific script. The character is also known as midpoint (in typography), Georgian comma and Greek middle dot (ano teleia).
The glyph is not a composition. Its width in East Asian texts is determined by its context. It can be displayed wide or narrow. In bidirectional text it acts as Other Neutral. When changing direction it is not mirrored. If its East Asian Width is “narrow”, U+00B7 forms a word with similar characters, which prevents a line break inside it. Otherwise it allows line breaks around it, except in some numeric contexts. The glyph can be confused with 207 other glyphs.
The CLDR project calls this character “middle dot” for use in screen reading software. It assigns these additional labels, e.g. for search in emoji pickers: dot, interpunct, middle, middot.
The Wikipedia has the following information about this codepoint:
An interpunct⟨·⟩, also known as an interpoint, middle dot, middot, centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in Classical Latin. (Word-separating spaces did not appear until some time between 600 and 800CE.) It appears in a variety of uses in some modern languages and is present in Unicode as U+00B7·MIDDLE DOT.
The multiplication dot (Unicode U+22C5⋅DOT OPERATOR) is frequently used in mathematical and scientific notation, and it may differ in appearance from the interpunct.