can you tell me which Operating System you are using? We already do that on macOS because it is common to keep applications running even if the window is closed there, but I'm not too familiar with the conventions on other operating systems regarding that.

That's on Linux (Ubuntu)

It also happens in Windows 10

On windows this is still the case.
We do get a little Tray-Icon on the bottom right, which is nice, but the moment we close the app itself the tray-icon and the widget disappear.
Idealy the app would close and TrayIcon and Widget would stay up, with the app running in the background and allowing me to reopen it with the trackIcon.

Seconding this. Minimising to tray, especially how the app is designed, would be key for solidtime on Windows.