I want to recess something into the wall between the studs. Problem is that it is approx. 1/4" longer than the gap between the studs. Is there a way to shave off this small amount of the existing stud (i.e. the "2" (1.5") side for about 2' length
You have several options here, and you're going to be stuck with some limitations as to which method you use. You can use a reciprocating saw, but as it was pointed out by @JACK, there's the danger of being unable to control the saw and cuttin all the way through to the other side.
You can use a hammer and chisel, but here you'll be shaking the entire wall and potentially cause some superficial damage that can cause some severe headaches if you have to match texture and/or paint. The reciprocating saw can also run this risk.
@Chris Butler also suggested a circular saw, which can work, but you're teetering on a very narrow edge and will cause a fair amount of mess.
My preference is to use a multitool if you have one available. If you don't have one, they're pretty inexpensive and very handy. This would allow you to do a plunge cut and you can use the fat edge of a speed square to help ensure you're keep to your 1/8" or 1/4" depth (1/8" if trimming both studs or 1/4" if only trimming 1 stud. These are also great in tight places.
You can try a combination of a laminate router, sharp utility knife, or chisel, but I wouldn't recommend that. I only mention it to try and cover all bases I can think of.
I would tackle that with a reciprocating saw. You could do two 1.75" cuts or one 3.5" cut depending on how comfortable you are with the saw. Just make sure you don't cut through the opposite wall. Cut your 1/4" cuts two feet apart and then cut out the strip between them. Make sure there's no wiring going through the stud.
I'd either do that old school with a hammer and chisel, or if I was feeling brave/foolhardy, with a circular saw with cut depth adjustment.
If you use a circular saw, check and set the cut depth to just minimally less than the depth of the stud that you're trimming, that way you can avoid damaging the wall and finish it off easily with a hammer/chisel.