Being On a Thing means two people are in the early, undefined stage of dating, more than friends but not yet officially together.
Saying you are on a thing, or that you have a thing going, signals an early connection that has not been defined. There is interest and maybe romance, but no label yet.
It is the stage just past flirting. You are seeing each other, talking regularly, and clearly more than friends, but neither of you has called it a relationship.
The phrase is deliberately vague, which can be its appeal. It keeps things light and avoids pressure during the get to know you period.
The downside is the same vagueness. A thing can drift into a situationship if neither person ever clarifies what they want.
It is a normal early phase. The question is whether it grows into something defined or stays in limbo, which usually depends on an honest conversation down the line.