The simplest assumption about the global properties of the Universe is that it looks the same outside the part that is observable to us, as it does inside. That is, we see a finite part of a Universe that is (probably) infinite in extend. If so, then the calculated age — which is finite — applies to all of the Universe, not just the observable part.
The age is calculated on the basis of the observed expansion rate, and the observed densities of the constituents of the Universe. It is possible to imagine a universe with the right mixture of constituents that has existed forever$^{dagger}$, but for our particular Universe, this just doesn't seem to be the case; it is ruled out by observations.
As a first-order approximation, you can simply take the age $t_mathrm{Uni}$ to be the reciprocal of the expansion rate $H_0 = 70,mathrm{km},mathrm{s}^{-1},mathrm{Mpc}^{-1} = 2times10^{-18}$ s. That is,
$$t_mathrm{Uni} sim frac{1}{H_0} = 14,mathrm{billion,years}.$$
However, this assumes that the Universe has been expanding at the same rate throughout its entire history, which is hasn't. More generally, the age is calculated from integrating (numerically except for simplified approximations) the Friedmann equation, yielding 13.819 billion years.
I should say that the calculated age is the time from the Big Bang till now. I guess the safest thing to say is that we don't know what happened the first tiny fraction of a second or so after creation, and in principle it could have existed before this instant, collapsed, and then re-expanded. But no observations I know of suggest this.
$^dagger$An example of a temporally infinite universe is one containing energy only in the form of a cosmological constant. In this case, the Friedmann equation reduces to $da/dt=aH_0$, with $a$ the scale factor ("size") of the universe, the solution of which is an exponential function with zero size only at $t = -infty$.
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