In Wikipedia, the age of the universe is defined as the "time elapsed since the Big Bang" while "time" links to "the cosmological time parameter of comoving coordinates" which itself links to "the elapsed time since the Big Bang according to a clock of a comoving observer", the latter being defined as "the only observer that will perceive the universe, including the cosmic microwave background radiation, to be isotropic".
Meanwhile, we also find in Wikipedia: "The theory of relativity does not allow the existence of absolute time because of the nonexistence of absolute simultaneity. Absolute simultaneity refers to the experimental establishment of coincidence of two or more events in time at different locations in space in a manner agreed upon by all observers in the universe."
While both makes sense to me, I feel like a contradiction between them in the sense that the age of the universe for the comoving observer located where and when an event occurs could be considered as the absolute time at which this event occurs. Indeed, such a definition of time would probably be impossible to implement in practice due to measurement uncertainties. However, it could be used in principle to define in which order two events actually occur in a way on which all possible observers should agree.
So, what did I miss and how can we reconcile these two points of view?