I believe this is by design.
Background processes (such as those belonging to extensions with the background permission) are only persisted until the user has explicitly terminated the browser (i.e. by clicking Exit within the ☰ menu). This is the expected behavior.
Closing windows, however, is not interpreted as an explicit request to terminate the application, which is why the background processes will persist in that scenario.
Consider the following from relevant documentation:
When any installed hosted app, packaged app, or extension has “background” permission, Chrome runs (invisibly) as soon as the user logs into their computer—before the user launches Chrome. The “background” permission also makes Chrome continue running (even after its last window is closed) until the user explicitly quits Chrome.