This character is a Nonspacing Mark and inherits its script property from the preceding character. The character is also known as short and Greek vrachy.
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 Nonspacing Mark. When changing direction it is not mirrored. U+0306 prohibits a line break before it. The glyph can be confused with 28 other glyphs.
The Wikipedia has the following information about this codepoint:
A breve (BREEV, less often BREV, neuter form of the Latin brevis "short, brief") is the diacritic mark ◌̆, shaped like the bottom half of a circle. As used in Ancient Greek, it is also called brachy, βραχύ. It resembles the caron (◌̌, the wedge or háček in Czech, mäkčeň in Slovak) but is rounded, in contrast to the angular tip of the caron. In many forms of Latin, ◌̆ is used for a shorter, softer variant of a vowel, such as "Ĭ", where the sound is nearly identical to the English /i/. (See: Latin IPA)