Can you access chrome:// pages from an extension?

When you're writing the manifest.json file, you have to specify matches for your content scripts. The http and https work fine, but if I try to include chrome://*/* or any variant of it, I get an error that I'm attempting to use an invalid scheme for my matches.

Is it not allowed?

By default you cannot run on a chrome:// url page.

However, there is an option in chrome://flags/#extensions-on-chrome-urls:

Extensions on chrome:// URLs (Mac, Windows, Linux, Chrome OS, Android)
Enables running extensions on chrome:// URLs, where extensions explicitly request this permission.

You still have to specify pages that your extension can run on and wildcards are not accepted - so you have to specify the full URL eg chrome://extensions/

The authorized schemes for matches are http, https, file, ftp.
Therefore, chrome is not a valid scheme.

Yes, it is not allowed. You can't link to them from hrefs on a webpage either.