Age in relativity is called "proper time".
In relativity, we have a four-dimensional "spacetime" which is independent of the objects in it (although in general relativity it is coupled to the objects in it). Spacetime is composed of events. Anything that happens in the history of the universe happens at some event or other.
Time is then one of four labels for an event (the other three being spatial labels, such as x,y,and z). The labeling can be done in different ways depending on your reference frame, so different observers may have different times for the same thing. However, time is not dependent on the object at an event. It depends on what event you're at and what reference frame you're in.
An object moving through spacetime will exist at many different events over the course of its history. The string of all the events an object visits is called its worldline. The age of an object is a function of the "length" of its world line. This age is called proper time; it is a natural parameter for the world line. Inside a given reference frame, the square of the differential of the proper time, $\textrm{d}\tau^2$ is some quadratic function of the differentials of coordinates, meaning
$$\textrm{d}\tau^2 = g_{00}\textrm{d}t^2 + g_{01}\textrm{d}t\textrm{d}x + \ldots + g_{33}\textrm{d}z^2$$
Once we know all the coordinates of the worldline and all the coefficients $g_{\mu\nu}$, we can find the proper time $\tau$ by integrating over the entire worldline. (It is not always possible to cover all of spacetime with one coordinate system, so we may need to do more than one integration.) The coefficients $g_{\mu\nu}$ depend on the reference frame used as well as the underlying geometry of spacetime. Once we know them, we can find the proper time, and therefore the age, of an object. So proper time is a property of an object's history. Time, as a coordinate, is not.
