I'm seeing the same behavior.
From the events log (apologies if someone already pointed this out), when I open the client, I see this:
2023-10-09 18:23:30.271 Application Quitting Indigo Server - received fast quit signal
then it goes through all the shutdown stuff (stopping plugins, etc etc.)
And then it gets to the end of shutdown and immediately restarts
2023-10-09 18:23:35.303 Application Quitting Indigo Server - disconnecting clients
2023-10-09 18:23:35.711 Application Quitting Indigo Server - unloading launchd plist
2023-10-09 18:23:35.719 Application Quitting Indigo Server - complete
2023-10-09 18:23:35.870 Application Starting Indigo Server version 2023.1.0 on macOS Version 11.7.10 (Build 20G1427)
I also ran a loop printing out the date and the ps output of the IndigoServer, and the pid/process start time remains the same until I open the client at which point a new instance is spawned.