is there a code stating how many sheathed electrical cable are allowed in a hole thru a stud?

There is nothing that specifically states there is a limit.

Keep the following in mind, on a 2x4 stud, you only have a 1" wide strip down the middle of the stud (wide side) that you can drill a hole in without using nail plates to protect the wires. A practical limit is 3 cables, 4 or more is possible, but now you're talking about doing some real work.Much easier to drill more holes.

If you do 5 cables into one hole and run through more than 2 studs, many inspectors will require de-rating (lowering the capacity of the cable). This can lead to other code related problems.

Tom