Agree with 90% of what
@ChrisG said. TruOil is really shiny stuff because it has some solids in it that are less than "oil". It is easier to use, hardens faster, and creates that "Perazzi" mirror finish. It also lets you do a job in 4-10 coats whereas a real oil finish might take 20-50 coats. (and the word coat is a misnomer, is a drop on the palm of the hand rubbed in every day for two months. (Trivia: Purdey's technique took >90 days. A coat a day for a week. A coat a week for a month. A coat a month for 3 months.) It also is a very modern look ala Roy Weatherby gloss rifles. I've tried to use TruOil and then to knock down the gloss on the final polish a bit. After a couple weeks of shooting the shotgun it burnished to gloss again wherever my hand and thumb were located making it look a bit wonky on a gun made in 1896. (A Lancaster Sidelock Single Trigger 12 bore)
There are finish kits out there that include the red alkanet root stain (or use vinegar and iron shavings if you want the alternative), slacum, and proper grain filler. If you want to restore it, you want to reapply the traditional finish techniques that will give you the right look. It appears from the photos its a German trade gun like a Simpson, a Sauer, or some other guild gun. They would have used a more subtle, subdued oil finish originally.