If the grandson's name is the same as the grandfather but not the father is the grandson considered the second?

If you name a child after its grandfather and the child and grandfather have the same surname and middle name (if any), then the child can be called "the second."

However if someone in your generation (a brother or cousin) is also named after the grandfather and has the same middle and surname, that person is "the second" and your child would be "the third."