Is atomic weight universal?
There is no "or".


Yes, we are advanced enough to calculate the abundance of elements throughout the universe, albeit somewhat hypothetically. (But then again, the estimates here on Earth are also largely indirect. The radius of Earth is 6400 km, and the deepest borehole pierced only 12 km.)
Yes, if you look at the Periodic table on the wall of your chemistry class, the atomic weights listed there are only based on the isotope abundances in our planet alone.


True, the "earthly" and "Universe-based" atomic weights rarely differ by much. One notable exception is argon (earthly product is mostly $\rm^{40}Ar$, cosmic version is mostly $\rm^{36}Ar$). More subtle discrepancies are numerous. They are well-known, documented, and routinely used to prove that a particular rock is a visitor from space (even if nobody saw it fall).

So it goes.