A Custom field ist not what I'm looking for, because of i want to filter for this field
You can filter by custom fields, check settings of custom field definitions

I don't see a reason at the moment to add a responsible user in addition to the existing "Assigned To" user.

Can you provide an example where the existing E-Mail notification configurations are not enough to get what you want?

External Partners use out Mantis for easier working together, so an Internal User is responsible for this Project (So this could happen that the Responsible User isnt assigned to the Project)

I've tried your Custom Field solution, but are there Notifications possible for this way?

And how could i sort this list (=users) or are there possibilities to list only the users who are able to be assigned?

You wrote

or are there possibilities to list only the users who are able to be assigned?

but you wrote before

So this could happen that the Responsible User isnt assigned to the Project

It seems that your feature request is not complete clear and also not needed for typical use cases of most MantisBT users.
Thus it does not make that much sense to bloat the MantisBT core with such kind of functionality.
Instead this is a typical request where a plugin makes sense.

Maybe this plugin is useful for you or might serve as a starting point for your own solution
https://www.mantisbt.org/wiki/doku.php/mantisbt:customermanagement

Do you agree that we should close this issue as "won't fix"?

yes, i've seen that this isn't possible to create a real "responsible Person" Field.

But is it possible to create a custom field where to choose only the assignable Users?
And further to set a Mail-Notification to this User?

But is it possible to create a custom field where to choose only the assignable Users?
Create a field with type enum and populate it by using a custom fucntion
see https://www.mantisbt.org/docs/master-1.2.x/en/administration_guide.html#ADMIN.CUSTOMIZE.CUSTOMFIELDS.DYNAMIC

And further to set a Mail-Notification to this User?
Write a plugin that catches event EVENT_NOTIFY_USER_INCLUDE
https://www.mantisbt.org/docs/master-1.2.x/en/developers.html#DEV.EVENTREF.NOTIFY

teekay12,

This is not a feature request for MantisBT that should be implemented in future versions, but more a candidate for a plugin.
I am therefore resolving this issue as "no change required".

Please use the forums, the mantisbt-help mailing list or IRC to get support on customizing and using MantisBT (refer to http://www.mantisbt.org/support.php for links and further details).