I have this same gun, bought it a year or so back. These guns, along with most brands of budget priced guns, used birch wood with a spray on finish that is a combination stain/polyurethane. I personally call it the mud finish. That's about how good it looks. The birch under it will look far better if you completely strip it bare, stain it with a stain color of your choice, then give it an oil finish as you would any other stock. I have done this many times on birch and I'm always amazed at what was hidden under all that mud.
Start by removing it from the gun, remove the buttplate, scuff the old finish with steel wool then use a paint stripper. I use citri-strip myself. A nice thick coat, let it sit overnight, then wipe off dried up crud and if needed, steel wool again and another coat. Rarely need more than that. If there is any checkering, use a toothbrush on it. Next is wash it off in hot water, dry with paper towels then let dry overnight.
Next is wood prep. If you have dings you can raise them with steam. If you have oil soaked areas you will need to use acetone to get the oil out. That process can take days or more because after washing with acetone and letting it dry, you should let it sit and see if more oil will rise to the surface. I don't sand much unless really needed with the exception of 400 before the final 0000 steel wool. If you do sand the buttstock, put the buttplate back on first so it will be even with the wood. Stay away from the metal contact area's as much as possible so you don't end with with a "proud" metal fit. Once I'm happy with it's condition, I'll rub it down gently with 400 or finer grit paper, followed by buffing it with 0000 steel wool. Next is a final wash with a wet acetone rag. Apply stain as desired, let dry, rub again with 0000 wool and if desired, more stain. Repeat as needed to get color where you want it. Bare birch when oiled is very light, so that's also an option if desired.
For final finish, you have numerous options and discussion on that usually carries on as much as the Ford vs Chevy vs Mopar debate. Depending on the gun, I use either BLO, raw linseed, Tung oil or raw Tung oil, Tru Oil or wood food oil. The key is patience. A nice hand rubbed oil finish doesn't happen overnight. It can take weeks of applying and drying and repeat, sometimes buffing lightly with 0000 wool and repeat etc. The biggest mistake most make is gobbing it on to thick thinking it will get done quicker. Rub in a thin coat, not thick. Sometimes I will dilute BLO or tung oil with mineral spirits, making it thinner to soak in better for th first few coats. Plan on it taking some time, be patient and you will eventually get it to where you want it. Bear in mind what type of gun you are working on. A utility grade gun can be made to look great with the right finish, but if you try to make your wood look like something on a custom grade high gloss Weatherby rifle then it will look as odd as no finish. Match your wood finish to the grade of the metal.