NASA is less than 60 years old. The science and equipment for measuring these phenomena is even younger.
There are known objects orbiting our Sun with orbits spanning hundreds of years.
The question of how it is known that this asteroid is interstellar is a fair one. It has not been answered, and it needs to be answered, else we are not talking about rigorous science here, we are instead talking tabloid science. The PR people of NASA often do that to create interest in the field. The photo they released is an ‘artist’s conception’ and their write-up may also be an ‘artist’s conception’.
“While originally classified as a comet, observations from ESO and elsewhere revealed no signs of cometary activity ... “
The quote is telling, the astronomers are not sure what they have.
Every object that is ‘slingshotted’ past our Sun immediately draws a curved trajectory because of the constant yet decreasing forces of the Sun’s gravitational fields. It is possible that an object attains a distance and speed sufficient to exit our Solar System, but the NASA write-up mentions variation in speed which is likely why it was classified as a comet.
No one is saying the astronomers who are touting this are wrong. But they need to explain and defend their assertions because that is what science is, a debate based on measurable, testable, repeatable, replicable quantifications.
Short answer: Its observed speed is greater than the escape velocity of the Solar System.
As of Nov. 20, 'Oumuamua is travelling about 38.3 km/s relative to the Sun. Its location is approximately 124 million miles (200 million kilometers) from Earth [...] though its outbound path is about 20 degrees above the plane of planets that orbit the Sun. The object passed Mars's orbit around Nov. 1 and will pass Jupiter's orbit in May of 2018.
This passage doesn't make it absolutely clear, unfortunately (the author is unnecessarily holding back known facts that would make the answer obvious), but it looks very likely that the author is saying that, at a distance of roughly 1 A.U. from the Sun, the object had a velocity which was a multiple of the escape velocity at the Earth's orbit (namely: ca. 42 km/s).
That means that the Sun's gravity will be unable to hold it.
The only other theoretically possible answer would be if the object had swung by (very, very closely) several gas / ice giants (Neptune, Uranus, Saturn, Jupiter) in succession, and thus gotten several "gravity assists" - something which NASA can accomplish only with modern computers and mid-course adjustments to the trajectories of its space probes.
Regards,