Flash Fiction is mostly seen as the bridge between Poetry and Fiction. In addition to Flash Fiction, there is Prose Poetry. Both Flash Fiction and Prose Poetry are short (one to two pages, usually), and often contain elements of the genre in which their name resembles. For example, Prose Poetry is written like prose, but possess many poetic elements (i.e. rhythm, rhyme, assonance, consonance, etc.) and Flash Fiction is short prose that contain fiction elements (i.e. plot, character, conflict, etc.).
Internet driven Flash Fiction like the Six Word Story and Twitterature usually lack elements of fiction, which is how Flash Fiction as its currently being published in it mags is beginning to stand out. Most lit mags only accept one piece of Flash Fiction per writer, so I haven't seen many instances of Flash Fiction collections with linking themes. However, a dear friend of mind is currently working on publishing his own Flash Fiction based off of fairy tales. So, they don't need to follow a theme, but they sometimes do. Regardless, Flash Fiction is much bigger than just social media based musing and typically stand out based on how well they incorporate elements of fiction.