Welcome to the S&W Forum. You have discovered a difficulty many of us encounter with family lore... our family members tend to remember what we think we remember, but it's not always complete or even correct. The revolver in question is definitely a 38 M&P, and it may well have been in your family since sometime "around the depression." As has been mentioned, the gun itself is older than the Depression Era, but during that era as now, guns frequently changed hands as payment for debts etc, and used guns were cheaper than new ones, so they were frequently attractive even if purchased at a store. It would be very nice and very satisfying (for me, at least) to ask around within the elder members of the family and friends and find out any additional information available. Sooner is better than later, as all of us are getting older and old folks' memories tend to get poorer with age, and we're passing away!
So far, the family lore easily agrees with the historical information that the serial number yields,
if we take as a given that the gun was bought used. If you find out great grandpa was a deputy sheriff in a small town where he had to provide his own sidearm, that would explain everything. If the ancestor in question was a moonshiner, he might have swapped a gallon of product for it.
My own grandfather in his later years warned me against looking too closely at our family tree, because as he said, I'd find a lot of horse thieves hanging from its limbs. Of course that made me look all the harder!
Again, welcome to the Forum. I hope you will enjoy being here and will find even more interest in your family heirloom.
Green Frog