Final answer:
The age of the universe is 13.8 billion years old based on measurements made in the frame of the expanding universe, which is the reference frame used by cosmologists.
Step-by-step explanation:
The theory of relativity does indeed teach us that the time interval between two events is frame-dependent. However, when we talk about the age of the universe, we are referring to the time interval as measured in the frame of the expanding universe. In other words, the 13.8 billion years is the age of the universe relative to an observer who is at rest with respect to the overall expansion of the universe.
While it is true that different observers moving relative to one another may measure different time intervals between the same events, the age of the universe is defined as the time that has elapsed since the Big Bang event, as measured by an observer who is at rest with respect to the overall expansion of the universe. This is the reference frame that cosmologists use when they talk about the age of the universe.
Therefore, regardless of who is asking the question, the age of the universe is considered to be 13.8 billion years old based on the measurements made in the frame of the expanding universe.