Hello again! Yes, generally speaking if the number on the frame and the barrel are the same, that is an indication that the barrel has not been replaced. However… for several years now there has been a small cottage industry in "boosting," "enhancing" or more pointedly simply faking the markings on higher priced lugers. I admit to having been taken in by at least one of these fakes. In the case of your luger, it's not clear why the person who refinished it left the serial number and the date but removed the other marks. It was very common in the 1950s and 1960s to have a firearm especially a luger refinished with that deep, dark, glossy finish that some folks associated with high-dollar firearms. That leads me to believe that your luger was refinished in the US some time during that era. However, that may not be the case. There is at least one other possible explanation. There were still a lot of GIs passing through Germany during that period and a lot of them wanted to come home with a souvenir such as a luger. I've even had GIs who were stationed in Germany in the 1950s and 1960s tell me that they could buy surplus lugers at their base PX. There were local gun shops that were more than happy to take a tired old luger and make it look bright and shiny again. In fact, some guys wanted them so shiny that they had their lugers chrome or nickel plated. This was done both in Europe and in the US. Ugh! In any event, in Europe the crucial part that is considered the core of the firearm and should have a serial number is the barrel while in the US the crucial part is the receiver. That may explain why the two serial numbers were left, but that is a real stretch and it doesn't explain why that other little number below the serial number was not removed. That little "8.83" number, if I'm reading it correctly in your pictures, is the bore diameter and was stamped there after assembly but before "power proof and shoot in" as outlined in point 26c of the official "Instructions on Marking the Pistol 08… 1910" as translated by Gortz and Bryans in their book German Small Arms Markings.
Now there are a couple of other points that may be of interest regarding your luger. First, it appears that it still has the stock attaching lug on the back of the grip frame. That is good. After WWII, a lot of folks were led to believe that any pistol which could be attached to a shoulder stock was illegal. Consequently, a lot of folks ground off the stock lugs from their lugers. We can't quite see all of it in your pictures, but it appears to be intact. Second, the magazine in your photos is a little dark but it appears to be for a much later, World War II era luger. If the base is black bakelite (plastic-like material) with a pin through the middle, it should be marked "fxo" along with a very small waffen eagle stamped on the metal near the base. This was the last style of magazine for German military lugers and in good, original condition is very desirable by itself. Third, if you would like to see what your luger would have looked like when it left the factory, go to the Rock Island Auction website and look at item number 812 in their upcoming December auction. This is a 1916 luger that has been professionally refinished and therefore looks pretty good. There are only a few photos, but one of them is of an original holster from the period.
Hope that I haven't bored you to death and you find some of this interesting. Thanks again for sharing.