When changing a flat, the time before last, I noticed the shift cable near the hub of my Nexus 8 was nearly completely broken. There was only a strand of two holding it together.

After a replacement cable was installed, when changing my next flat, the cable looked good, but I just noticed when changing yet another flat, the cable has stared to break again.

This time there is just a strand or two that's broken, but it's on it's way.

Is this normal? Do shifting cables break down this quickly?

How quickly?

Too quickly.

I had the nexus hub put in about 3 years ago and the cable replaced 2 months ago.

I'm not too sure when the previous cable had started to break down, so maybe it had been breaking down a strand or two for a while. I'll keep an eye on it. Maybe it'll last another year or two but it seems too fast to break.

I think in over a couple of decades of riding, I only had one other shifting cable break before. Of course, that was with a conventional derailuer system,

Consider upping your tool level, so you can Solder that portion of the cable
in advance of it being cut.. so the cable will not fray after the cut..
Zn (die drawn) cables are easier to solder than SS ones..

Where is the cable breaking down? A photo would be helpful to diagnose what is happening.

right at 12 o'clock

(The bike is resting upside down )

so that's it? No one knows if this type of wear is usual?

Disclaimer: I cannot see the picture(s)

On my Alfine hub (same cable attachment as Nexus, I believe) the cable has become frayed at the clamping nut on the loose end almost every singe time I have assembled it. If it doesn't fray when I assemble it, it frays the first or second time I remove the calbe to get the wheel off. In my experieice with Nexus hubs, I can recall a few strands breaking on the other side of the nut (the tensioned side) on assembly. I think it is an issue with the clamping nut, and you need to be 125% gentle with it during assembly to avoid any fraying or broken strands.

Not usual -- 4500+ mi on the original cable on a Nexus 8 system and nary a fray like you are showing.

Are you routing the cable through the hub cassette joint correctly?

Yup, although I suppose anything is possible.

It's a pretty simple routing and if it isn't done properly, it wouldn't shift right, correct?

Keeping it a bit cleaner and trying some dry lube might help as well. Instead of dirt and sand grinding away at it all the time the lube would fill in the cracks and prevent that.

Worth a shot after you re-replace the cable.

It wouldn't shift correct initially, but you could dial it in to get some kind of shifting. If you're familiar with it, probably not the issue, but the only one I can think of beyond dirt/use that might be doing that kind of damage to the cable. Radius at the hub cassette joint is bigger than Shimano STI shifters with the side entry, where cable fraying can be an issue.