For what I can tell, after the clickbaits and the sensationalisms, there is an actual unexplained discrepancy in the values of the Hubble constant measured with different methods. This means that something is not right in one of the methods (perhaps both), or that we are missing something else.

And yes, the Hubble constant is closely related to the age of the universe, so it's basically the same issue.

This is actually a good thing, because the "crisis" can be the first clue to discovering some new exciting science! Just look at how the "ultraviolet catastrophe" kickstarted the field of quantum mechanics a century ago

The Hubble Tension is certainly real. The Hubble Constant can be estimated from a number of completely independent astrophysical phenomena. There is a significant difference in the value computed from 'local' phenomena, and distant phenomena. It is often referred to as the "5 sigma" tension, because that is the statistical significance of the disagreement. This has been know long before James Webb, but as the article says, these observations just lowered the uncertainty on one of the probes. But we were fairly certain already that this tension is real.

Whether it's a crisis or not is up to the individual. Things not agreeing in science

especially astrophysics/cosmology

is just part of the process. I don't know anyone that is 'worried', so much as looking for ways to solve the problem.

Dr Becky will get you fixed up:

https://youtube.com/playlist?list=PLd19WvC9yqUf5TRqYoMYxEwjT6JIDW4Zn

Go in chronological order if you want the full picture, but it's not really necessary.

I'd describe it as a faux-crisis. It's not like astronomy as a field is going to end if we don't get it sorted by next week. It is annoying though.