How can I enable Android Debugging from a root shell on the phone? I have a root shell on the phone.
However, no adb service seems running, i.e. the port 5555 is on the device not open1:
$ telnet 1.2.3.4 5555
Trying 1.2.3.4...
telnet: Unable to connect to remote host: Connection refused

I.e. something what should listen on adb connections, does not run on my Android.
Can I somehow start it from a root shell (on the Android)?
I have Android 10.
1Telnet only serves here port testing purpose, obviously adbd will be connected by adb.
Run:
settings put global adb_enabled 1

You can also start adbd service by executing start adbd or setprop ctl.start adbd. But I prefer the settings way because it properly goes through Android framework.
PS:
For clarification, adbd listens on port 5555 only if ADB over Network is enabled (from Developer Options, or by running setprop service.adb.tcp.port 5555 on device, or by running adb tcpip 5555 on PC). In USB mode (which can be switched to by running adb usb on PC) adbd is not listening on any network port. Some more related details in this answer.