An upgraded site should not have the libraries/cms/router directory, the files were moved to libraries/src/Router. This sounds like either a plugin conflict or the upgrade had issues.

Just checked. Although uninstalling the jlrouter plugin 'sucessfully' according to the Joomla message, I have a different versions of router.php in both locations. If I disable the one in libraries/cms/router, the two error notices go away.
Question: Does the built-in router in 3.8Beta have configuration options, like those in the jlrouter plugin somewhere? Don't see them anywhere. In 3.8Beta I get menu item ids for category blog and category list links in the SEF url.

It sounds like this JRouter plugin doesn't have a correct uninstall method then.

The core 3.8 new router can be enabled per component options -> integration

Tested again with J3.8 Beta2. This time the update worked and the new router seems to work too.

(But I admit, that I like the new router as a plugin better, because it provides a central place for the setting of all components. I'm using the new router plugin on several websites since 5 months without a single issue)

When you guys release J3.8 stable, do you still intend to label the new router 'Experimental'?

My Git staging/production environment uses zipping extraction from the Git to update the file structure after deploying; so if you're going to 3.8 and you see a 500 error with:

Notice: Constant JROUTER_MODE_RAW already defined in ...\libraries\cms\router\router.php on line 17
Notice: Constant JROUTER_MODE_SEF already defined in ...\libraries\cms\router\router.php on line 24

in your error logs for Nginx you need to drop your current development libraries folder and replace it with the libraries folder from your staging area. This cleared a 500 level error from our Joomla environment and has worked every time (on two Joomla instances so far).

@reboot-sequence please keep noted that simple file overwrite (from a file or a FTP copy job) isn´t a supported upgrade option since joomla 3.x... If you do only overwrite the files you will get a wide range of issues as seen here for example as some background tasks aren´t running. The only supported way is to use the build in upgrade option from joomla. We used the same approach as you in the past and where advised by the joomla core team to stop the unsupported approach as it will lead to issues and is no longer supported (see here).