Much confusion over the years about cartridge designations. Prior to the 1950s most major firearms companies (Colt, S&W, Savage, others) manufactured guns primarily for their own proprietary cartridges. Those who offered guns chambered for a popular cartridge developed by another company were reluctant to allow the competitor's name to be displayed on their firearms.
.38 S&W Special is one example. Developed by Smith & Wesson as a military and police cartridge having greater power and powder capacity than earlier .38 caliber cartridges (.38 S&W, .38 Long Colt, etc). For some reason (probably potential military contracts) the development consisted of a lengthened .38 Colt Long case, which was then the standard US military handgun cartridge.
Colt and other makers quickly jumped on board with revolvers chambered for the new cartridge, but adopted the simplified moniker of .38 Special so as not to offer a nod to Smith & Wesson.
If the OP's revolver won't accept SAAMI-spec .38 Special ammunition the most likely reason is chambers in serious need of cleaning and removal of accumulated carbon and leading deposits. Another possibility would be damaged chambers, such as rust or pitting from poor maintenance.
More remote possibilities might include a revolver with a non-standard cylinder installed (.38 S&W cylinder from another M&P or Victory Model in that chambering), or defective ammunition (such as reloads with improperly sized cases, seriously oversized bullets, deformed case mouths from improper crimping), or perhaps a damaged extractor star installed.
I would start with a thorough cleaning and inspection of the chambers. I have found several older revolvers that had never been properly cleaned, but responded nicely to simple bronze bore brush, good penetrating solvent, and normal maintenance procedures.
Beyond that, an experienced gunsmith is likely to identify any mechanical problems with a few minutes of inspection.