Add the depth of the wood backing and the depth of the drywall (or paneling), then add the depth you want the screw to be in the stud. 1 1/2 inches in the stud is very unlikely to hit any electrical.

1" of screw engagement in the stud should be fine as long as you don't intend to do chin-ups from that item (whatever it is). In any case, the likelihood of drilling in the one location in the wall where a cable may be and hitting that spot exactly is pretty remote. I've never had that happen in decades of putting fasteners in walls. However, when going deeper than an inch, I always drill very slowly and make sure I'm continuing to get resistance from a stud: I'd stop if I felt the resistance dropped off as it could possibly indicate I'm drilling into the side of a hole in a stud where a cable might be.

Makes sense, it's an item that I am manufacturing for cats. I chose screws that get 1" engagement with the stud, hopefully that's enough.

The pullout force for a #6 wood screw with just 3/4" penetration is about 300lbs, and a #8 wood screw is about 350 lbs. Unless you drilled too large a pilot hole or the cats have been pumping iron, you should be fine ;-)