This issue should be fixed by 10.2.5 (it was one of the reasons for doing that release). You have have a cache issue, try to "hard reload and clear cache" or if the site is behind something that may cache resources (like cloudflare), you may want to also clear cache there.

Thanks for the pointer. After doing the hard refresh the visual problem with the upgrade page went away, so I was able to initiate the upgrade to 10.03.03.  After the upgrade was complete, the issue I was describing with my skin.css getting overriden also seems to be fixed.  So my portal is looking normal again.

Should I approach my upgrade in the production environment any differently?  Or is it just a matter of doing the hard refresh.  I'm still confused by 10.02.05 having display issues above and beyond just the persona bar.

Well, you should not have needed to clear the cache so it is hard to really tell you if you will need to do that or not on other hosting instances. But if you see the styles broke right after an upgrade, very often it is because of some level of caching.

Just to add and confirm Daniel's answers, the issues priority to DNN 10.02.05 that allowed some previously cached CSS to survive the upgrade was fixed. So, if you upgrade directly to 10.02.05 (from any version between 9.13.09 and 10.02.04), you should not experience the problem.

Further, fully hard-clearing the browser's cache and data for the site/URL should also fix the problem.