Please attach a net-internals log per these instructions. Thanks!
https://dev.chromium.org/for-testers/providing-network-details

This file is recorded when doing the following steps:

Go to the site (which points via hosts file to 127.0.0.1): E.g. https://www.my-local-dev-page.com
Get an error: NET::ERR_CERT_COMMON_NAME_INVALID
I click continue
I click on the red lock in the top left corner inside the certificate popup I choose "export to file"
I choose "DER-codiert-binär X.509 (.CER)"
I click next and close chrome
I import this CER file both via windows user certificates and also in the chrome settings in the ssl path of the root authorities

Deleted: chromelog_before_restart.txt 3.8 MB

chromelog_before_restart.txt 3.8 MB View Download

This file is recorded when doing the last steps:

I open up chrome and the error comes again
If I click again "continue", it seems that somehow it is working, but after a few requests, it is falling back and the error comes again.

Deleted: chromelog_after_restart.txt 2.6 MB

chromelog_after_restart.txt 2.6 MB View Download

ERR_CERT_COMMON_NAME_INVALID is not overridden by importing a root certificate. Or any certificate.

Thanks for your answer!

I am stunned. How can I then develop locally with a self signed certificate? Am I doing something completely wrong?

What is the way or is this not possible?

You need to generate your certificates with the right name.

Thanks, that was the case. The CN was wrong! Sorry, this topic can be closed!