How do we measure the age of the universe if time is relative to the observer? What is the reference frame we use to measure the age of the universe?
Cosmologists use the “comoving frame”, which is the unique reference frame at each point in which the universe appears isotropic. For example, if you measure the temperature of the cosmic microwave background to be 2.725 K in every direction, you are in a comoving frame. Observers moving relative to this frame see a hotter CMB in the direction of their motion.
So the age of the universe is, in principle, the age measured by a clock in a comoving frame. Of course, it is really a calculated age, not a directly-observed time on a clock. We calculate the age of the universe by measuring how fast the universe is expanding, and by using General Relativity to extrapolate back in time to the start of the expansion, the Big Bang.