When using Docker build and publish step with a docker tool configured globally:

• there is no way to choose which Docker Tool installation to use
• not any tool is ever installed

The plugin has no references to the DockerTool whatsoever and uses hard-coded docker binay name instead of refering to DockerTool.getExecutable at https://github.com/jenkinsci/docker-commons-plugin/blob/0160e2b3ea862e9476c5d9be2d6cf66379ca6df8/src/main/java/org/jenkinsci/plugins/docker/commons/tools/DockerTool.java#L73

See https://github.com/jenkinsci/docker-build-publish-plugin/blob/master/src/main/java/com/cloudbees/dockerpublish/DockerBuilder.java#L344 for example