Almost certainly some stain, if any is used, will be left. It's tough to wash that stuff out with any paint and varnish remover.
But most of the better wood for gunstocks is not stained. It's that colour naturally. And in fact staining many of the woods used is pretty much like putting mud on it and hides the grain. The only time you want to use a stain is if the wood is something cheap and lacking in grain contrast. In such cases stain can help the appearance by increasing the contrast between the soft and harder parts of the wood. To do this without looking muddy and hiding more than it helps you need to put it on then rub away most of it. And that means it's well into the wood and you can't get in there to remove all of it.
If you don't need to use a stain and just a varnish or oil finish brings out a nice colour then you can strip and re-do it as many times as you like. Some light sanding will likely be needed but if you do go lightly it won't remove any significant amount of wood if it's just a light sanding to remove the last bits of varnish.
If you're asking because you want to try different stain colours then that's different. Over time the residual from the first few will affect the total as you get into the third and fourth and more attempts. You're better off to work with the stain colours, if you have to use stain at all, in one go. If you start with a lighter stain than you think you want and it's too light you can wash much of it away with low odor paint thinner right away. Then mix in some darker brown or darker red and brown to modify the too light stain and apply then wipe off again. If it's still too light add a little more of the darkening stains to the mixture and try again.
You'll find that the paint thinner doesn't wash away all of it. And going from something too dark to a lighter look gets hard in a hurry. So it's better to start light and work up.... unless you want a higher contrast to the softer and more porous wood that is.
In any case don't use the stain as a paint. Use it as a stain. And that means lots of paper towels or rags that you can discard rubbed vigorously to remove as much as you can. And the rubbing off of most of the stain typically means that you want to start out darker than you think you want. If after rubbing the excess off firmly if it's still too dark you can wash some of it off with a rag lightly dampened with the thinner.
With all this thinner and oil based stain do be sure to wear solvent resistant gloves. That stuff isn't good for your skin or internal organs.
