I think it's more like how it's Pope Francis and not Pope Francis I, since he's the first Pope Francis, you don't add the number. We will only do that after the fact, i.e. when another pope chooses the name Francis.
So grandparent's point isn't that it's off by one, but that the first name is inaccurate for the time it was rewarded. Although, considering we are viewing it _after the fact_, i.e. now that more dogs are called the same, I think it's fair to add the "I" at the end.
No, they only allow 6 characters for the number of the dog. To write 38 in Roman numerals requires 7 characters. So that's why they only allow 37 of the same name.
