MacOS – Does Touch Bar support background / system wide applications

The Touch Bar is not meant to show background and/or system wide notifications.

Apple has put out a document called macOS Human Interface Guidelines which is intended to communicate to developers how UI elements should be implemented; there's a section specifically about how the touch bar should function.

Under the Design for the Touch Bar Section, it provides guidelines as to how the Touch Bar should be utilized within the context of an App.

Design a contextual experience. Make the Touch Bar relevant to the current context on the main screen. Identify the different contexts within your app, and consider how you can expose varying levels of functionality based on how your app is used.

And more specifically to your question, because "background/system wide applications" would provide notifications and it's UI and to the screen (via Notifications most likely), not through the keyboard:

Use the Touch Bar as an extension of the keyboard and trackpad, not as a display. Although technically it’s a screen, the Touch Bar functions as an input device, not a secondary display. The user may glance at the Touch Bar to locate or use a control, but their primary focus is the main screen. The Touch Bar shouldn’t display alerts, messages, scrolling content, static content, or anything else that commands the user’s attention or distracts from their work on the main screen.

(emphasis mine)