A year is a well defined thing.
I can see...

In astronomy, a Julian year (symbol: a) is a unit of measurement of time defined as exactly 365.25 days of 86400 SI seconds each.
Let's talk about seconds instead of years then, seeing as a year is ambiguous in its own right.

Here's the definition of a second...

It is quantitatively defined in terms of exactly 9,192,631,770 periods of a certain frequency of radiation from the caesium atom: a so-called atomic clock.
Two people will observe the time it takes for such a frequency of radiation differently, based on their differing velocities.

So, a second is ill defined.

Relativity is a beast, and it shits on our concept of time. When we apply time dilation to universal scales, we're not talking about negligible amounts like when we talk of horse lengths. We could be talking infinite amounts. To a photon originating from the big bang that is constantly in motion at c, the universe is exactly zero seconds old. To something that is not in motion at all, maybe it's infinitely old. Motion, of course, is another thing that's relative. Nothing within the universe is motionless, so nothing within the universe is infinitely old.

But maybe the universe itself can be considered motionless, because there is nothing that it can be in motion relative to.

So, I conclude that the universe is somewhere between zero and infinite years old, depending on the frame of reference.