This character is a Math Symbol and is commonly used, that is, in no specific script. The character is also known as existential quantifier.
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 mirrored. If its East Asian Width is “narrow”, U+2203 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 one other glyph.
The CLDR project calls this character “there exists” for use in screen reading software.
The Wikipedia has the following information about this codepoint:
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain. Some sources use the term existentialization to refer to existential quantification.
Quantification in general is covered in the article on quantification (logic). The existential quantifier is encoded as U+2203∃THERE EXISTS in Unicode, and as exists in LaTeX and related formula editors.