Is the commonly accepted Universe age of 13.8B years really accurate?

The significance of the tension is not that the age measurement is off by much, but rather such a tension strongly suggests that we have not yet correctly understood all the relevant physics. So the most likely outcome is that the physics gets understood better and all the age measurements turn out to be roughly right, just not quite as accurate as has been estimated on the basis of the models used so far.

I am not working directly in this area, so I cannot give an estimate of what the range of uncertainty might be. But it is clear from the publications that the Planck collaboration published measured values of cosmological parameters based on assumptions which included, among others: that if there is any spatial curvature, it is negligible; that if there is any local flow or other such inhomogeneities in the cosmic fluid, then its effects are negligible; that the dark energy has the form of a cosmological constant; various things to do with neutrinos and lensing which I don't claim to know much about. As soon as one allows that things like this cannot necessarily be assumed, one will get bigger error margins. So it seems to me that the error margins published by them are indeed a bit misleading, but I am not able to assess what would be a fairer statement. I note that Di Valentino, Melchiorri and Silk (2015) published a comparison of a 6-parameter model and a 12-parameter model. The error margin in the baryon density parameter in the 12-parameter model is twice what it is in the 6-parameter model. The error margin in the cosmological constant in the 12-parameter model is about 4 times larger than in the 6-parameter model. The error margin in the Hubble parameter in the 12-parameter model is $4.8$ times larger than in the 6-parameter model. Yes that's $4.8$ times.

So this indicates the kind of thing that can happen. More work has been done since then, of course.