I am not sure about Linux.
I would think that this is a problem that would greatly affect older systems (like all the banks' mainframes... yikes!)
[0] https://undeadly.org/cgi?action=article;sid=20130813072244
Mainframes don't run Unix in the way we think of it, and the applications they run store timestamps in a different format (hence the panic about Y2K back in the day). I'm by no means an expert, but I think these people don't need to worry about Y2K38.
The financial world has not collapsed so far, so they must have found some way to deal with that. But I'm not sure I really want to know what "solution" they went for.
That’s a big project with a lot of players and we won’t know if they all did their job right until the clock rolls over.