Hello and Welcome:
No one else seems to be willing to take a stab at this one, so I will step forward and volunteer to embarrass myself. Let be begin with my standard disclaimer that I am NOT an expert on FN's or Mausers. I enjoy and appreciate them and have had occasion to examine a few over the years. I can offer a few observations with regard to your rifle and the family story that accompanies it. First, based on your pictures, the "FABRIQUE NATIONALE" and "HERSTAL-LIEGE" appeared originally on the Belgium Model 1889 rifle and carbine. A photo of the receiver appears In Robert Ball's book Mauser Military Rifles of the World. However, when you look at pictures of the Model 1889, one of the first things that you notice is that it has an extended magazine but your rifle has a magazine floor plate that is flush with the bottom of the stock. The explanation is that the Model 1889 was "officially" modified at least twice by the Belgians--first in 1916 and then in 1936 and reportedly remained in service in some places until after World War II. FN had been working on the development and production of what was to become the FN49, but that was interrupted by the war, so the older models remained in service albeit often with modifications. Your rifle rifle appears to fit that pattern. That leads us to the second point about it being "an old service rifle that he got issued in the navy." That would be .... possible but very unlikely. The appearance of the Mauser Model 1889 coincided with a couple of events. First, the Model 1889 was so new and impressive and the facilities at FN were so limited that a US company, Hopkins and Allen of Norwich, Connecticut, received a contract to manufacture the rifles. The ones made by H&A are marked accordingly and yours is not. However, the US was seriously interested in replacing the old Trapdoor Springfield and the US trials did include the Model 1889. The fact that your rifle does not have the extended magazine of early examples suggests that it is one of the later modifications, not one of the rifles from any of the earlier trials.
In looking at your rifle, there are two other points that sort of jump out. First, your rifle has a bent bolt handle but does not have the stock indentation for the bolt handle. This configuration was seen on the 1889 carbine, but the carbine also had the extended magazine. Second, the provisions for mounting a scope appear to be more modern than would have been found on an early Mauser. Almost universally, the early Mauser scope mounts were "turret mounts" or sometimes called "claw mounts". The manner in which the scope was mounted on this example resembles the method used in the US from the late 1930s through the early 1960s. The two big challenges to mounting a scope on a Mauser where a) mounting it so that the bolt would still operate without hitting the scope and b) the safety would still be usable. Your rifles appears to have the original safety, so that wasn't a problem and the bent bolt handle would have helped with that issue. When the Germans started using the side mounted scopes on their 98s, they generally took out small piece of the stock and changed to the low profile safety. Your stock doesn't appear to have been altered and, once again, it has the original safety. After taking all of that into consideration, the four holes in the receiver rail are more commonly found on personal firearms from the 1930s through the 1960s. One final concern is the caliber. The original Model 1889 was generally in 7.65x53mm Argentine. Would that caliber be likely to have a "following" or a use of any kind in the US navy? Probably not.
So, how do we reconcile all of this limited insight with the family story? Well, a lot of souvenirs came home during the Second World War, which is probably when your "pawpaw's" brother was in the navy. The navy guys, especially those who were largely confined to shipboard duty, often felt a little left out. They didn't have as many opportunities as the average "ground pounder" to get the "good stuff" to send back home. So... they made do with whatever came their way. Things found their way on board either when the ship was in port or when some of the guys who had been ashore returned. The navy guys had a couple of things that the "ground pounders" didn't have and that was time and equipment. The navy guys had the time to alter and/or customize their souvenirs and if they were on board a large ship, they had access to a machine shop that could do significant work, such as mounting a scope.
The only other point that I have to offer is that the US government did purchase some Mausers to give to the insurgents in the Philippines in the months leading up to the Spanish-American War in 1898. It was supposed to be one of those "secret" operations to undermine the Spanish in the islands so we didn't want to give them anything that would point a finger back at the US if it was captured. These would have had the extended magazine, and probably have been in pretty rough shape by the time any sailor picked one up as a souvenir in the last days of World War II, but is it possible....
This is the best that I can offer. It's not very much. and I'm certainly not very satisfied with it. I hope that someone more knowledgable offers a better explanation so we have better insight into the history of this rifle. Once again, thanks for sharing.