Is there a preference to set so Bridge 2023 does not completely close down when exing out? InDesign and Photoshop do not close out completely.
At this point in time, no.
When an application is written to only have one window, when that window is closed, there's nothing left for the application to do other than to Quit. (For the record, Bridge is not the ONLY application that does this. The biggest issue here is that it is a new dynamic to the application that is catching people off guard.)
Hopefully, due to the increased complaints about this, something might be done in the future.
Apple's yellow button to Minimize; or Cmd H to Hide Bridge could do what you want?