Project column not showing team ownership

When working across several teams, it’s inevitable that each team will end up with the same project name. For instance, we have an “Administrative” project for each Team while each team represents a different project. The issue comes when you are assigned tasks from the different teams, the task will show in your inbox with the “Administrative” project, however you have no clue for which team this is so it’s impossible to use this intuitively.
Is there any way to show the teams alongside the project.

To go around this issue we sometimes includes the team acronym or name in the project name.