Calling something young or old is normally done in reference to some special age. For things that have limited lifespans it is usually the average life expectancy (old people are people who are close to or beyond the life expectancy). For things that can potentially remain arbitrarily long we compare to some typical timescale of their evolution, in particular the longest dynamical timescale (one might call a piece of a decaying radioactive element "old" when it has been around more than a few half-lives).
Since the current $\Lambda$CDM consensus model of the universe has an unlimited long future we need to find a good reference timescale.
We are living in the stelliferous era, when stars form and shine. This lasts from some hundred million years after the big bang until $10^{12}$ to $10^{14}$ years in the future. Many people implicitly assume a biocentric perspective thinking that life is only possible during this era and that the end of stars would be the end of the universe, at least the end of anything interesting. By this standard we are in the early part of the stelliferous, so it is a young universe.
However, as discussed in (Adams & Laughlin 1997) there is a lot more happening in later eras. My own research indicates that the degenerate era $\sim 10^{14}$ to $\sim 10^{33}$ years might be quite habitable for technological life. Even if one disregards life, in terms of change of structures and matter composition there are significant future changes: a separation of gravitationally bound objects in $\sim 10^{11}$ years, merger processes evolving galactic contents into denser forms, the dissolution of galaxies in $\sim 10^{20}$ years, proton decay removing baryonic matter in $\sim 10^{33}$ years or more, black hole decay in $\sim 10^{60}$ to $10^{100}$ years... basically all of these timescales make our present moment early and the universe young.
To claim the universe is old one needs to have a reference timescale that is fairly short. But even the particle physicist saying that what truly matters is the formation of stable particles and hence we should use baryogenesis during the big bang as the timescale has to admit that the stelliferous era involves a lot of particle conversion, and the degenerate era will involve a lot of conversion of protons and electrons into neutrons inside neutron stars (not to mention proton decay). I think it is hard to argue for any early timescale in any principled way. So we are living in a young, evolving universe.