Loaded question.
All share the same bullet .355/.356
You can shoot the 9mm Largo in a 9x23 but due to the higher pressure of the
9x23 is not a good idea to shoot a 9x23 out of a Barrel marked 9mm Largo.
38 super is a straight walled case, the 9x23 has a similar taper as a 9x19.
Therefore the 9x23 would be sloppy at the case head and tight at the base.
All this being said it is doubtful that a Gunsmith would recommend shooting the
9x23 out of a 38 Superb Barrel, however the round was originally developed to be shot out of a non ramped 38 super barrel as a solution to help the IPSC shooters make Major Classification. Those shooters will tell you if the 9x23 will drop into the
38 Super barrel via the plunk test, it is safe to shoot it. My 38SUPER Wilson X-Tac,
38Super Les Baer Mono HW, and DW Blue Valor I converted to 38 Superb using a Dan Wesson PM38 barrel all passed the plunk test and feed and fire the 9x23 without any high pressure indications.
The secret to the 9x23 .357 magnum type ballistics is derived from the thicker case web. The case was designed to be safe when fired from a Colt 38 Super non ramped (non supported) barrel without blowing out or bulging at the base. Nolan 9x23 and 38 Super barrel chambers are notorious for being tight and will more than likely need to be reamed with a 9x23 chamber reamer to reliably chamber the 9x23.
If the round will chamber without resistance in a modern supported
38 Super barrel, then you should be good to go. The slide velocity and recoil from firing the 9x23 will require a heavier recoil spring and heavy duty FP spring like a 1911 in 10mm. It may also benefit from the use of a flat FP stop to delay rearward slide movement just long enough for the pressure to peak and start to decrease.
Before you go out and buy a bunch of 9x23 and start blasting away, please
do your home work. The SD capability of 9x23 is impressive. Winchester whitebox 124 gr JFP has a muzzle velocity of in excess of 1550 fps. However commercial ammo is no longer produced and if you find it prepare to pay $30/box of 50 rounds.
Starline does however make brass for the 9x23 so if you reload....
One more thing.
Don't be tempted to shoot the 9x23 out of a barrel shorter than 5".
I believe
@Joe C recommended that the 9x23 creates significant issues if fired
from a Commander length gun.
Hope this was too long a post!