This plugin cannot be uninstalled - clicking the "remove" button in the UI does nothing to remove it - it says "pending"  but after a restart the plugin is still there and active.

The plugin can be deactivated, but this does not prevent the notifications of an "update" from being shown.

If physically removing the files, they are downloaded on the next Jenkins restart. How do I get rid of a plugin I do not need (and which I never installed)?

The jdk-tool plugin is an implied dependency of a large number of plugins because it was once bundled inside Jenkins core. A detailed description of implied plugin dependencies and how to resolve them is included in JENKINS-70301. The Jenkins plugins site describes implied dependencies by saying:

Features are sometimes detached (or split off) from Jenkins core and moved into a plugin. Many plugins, like Subversion or JUnit, started as features of Jenkins core.

Plugins that depend on a Jenkins core version before such a plugin was detached from core may or may not actually use any of its features. To ensure that plugins don't break whenever functionality they depend on is detached from Jenkins core, it is considered to have a dependency on the detached plugin if it declares a dependency on a version of Jenkins core before the split. Since that dependency to the detached plugin is not explicitly specified, it is implied.

Plugins that don't regularly update which Jenkins core version they depend on will accumulate implied dependencies over time.

The WMI Windows agent plugin was previously a widely used implied plugin dependency that has been successfully removed. https://github.com/jenkinsci/jenkins/pull/7568 describes that process.