We know how fast it is moving and how far from the Sun it is. If its velocity is greater than the escape velocity of the Sun at its current distance, it can't have gained that velocity just from its orbit around the Sun (because by definition an object in an [elliptical] orbit is traveling more slowly than escape velocity).

It is possible for a Solar System comet to be perturbed by other effects (like a close passage with Jupiter) into an escape orbit. But in those cases, the speed above escape velocity is small, and the orbit barely escapes. 3I/ATLAS is moving much, much faster than that, too fast for within-the-solar-system effects to explain it. It must therefore be interstellar.

This object was already discovered and known to have an hyperbolic (uncaptured) orbit.

Short answer, yes. But it's many frames, and over a time span of many nights and now weeks.