Like you, I walk and shoot my guns, (mostly fine walnut furniture) in the woods, and am obsessive about keepimg them in perfect condition. If I scratch or dent one, it gets repaired soon after. I use Maccari Royal Lundon Oil, which is easy to spot repair, and never know it happened.
That looks like an oil finish to me. Which is my method for refinishing all stocks.
If it is just a rubbed oil finish, which many FX and other European models often use on the higher grade guns, and not a satin urethane or similar hard finishes, you can steam without problem, as long as you head the advise of just short duration iron contact. Approx 10 seconds is my typical max, and repeat.
I think you might be able to make the small marks disappear, and reduce the larger ones by 50% or so.
Just my 2¢
Good luck,
Have fun