GitLab-CI: How to build docker image from other repository?

You can use

git clone https://gitlab.example.com/tanuki/awesome_project.git

to clone the remote repo. If the repo is not publicly accessible, you can create a deploy token to do this and use git clone with the deploy token.

git clone https://<username>:<deploy_token>@gitlab.example.com/tanuki/awesome_project.git

see: https://docs.gitlab.com/ce/user/project/deploy_tokens/#git-clone-a-repository

I would like to have all my dockerfiles in one repository, instead of having them in the seperate projects. Therefore I have created the project "conductor". All my .gitlab-ci stages are also specified in "conductor". In the actual project I only use the include command.

It looks like a good usecase to define the generic stage/template in the common repository and to to include them in the users repositories via the include keyword.
Look at the include documentation :

include Use the include keyword to include external YAML files in your CI/CD configuration. You can break down one long gitlab-ci.yml into multiple files to increase readability, or reduce duplication of the same configuration in multiple places.

From user git repositories you could include the common template in the .gitlab-ci.yml such as :

include:
  - project: 'my-group/my-common-project'
    file: '/templates/.gitlab-ci-docker-build-template.yml'

And of course, reference the common stage(s) you need to use in it.
Note that the common templates are not ridid. User/consummer repositories may override some things. For example variables are overridable.

The actual project could add conductor as a Git submodule (git submodule add.
In conductor, you would store your Dockerfile, one per actual project.

That way, your actual project can use

docker build -f conductor/dockerfile.actual