There's more to this than I imagined. From Wikipedia:
In the United States the most common name suffixes are senior and junior, which are written with a capital first letter (Sr. and Jr.) with or without an interceding comma. In Britain these are rarer, but when they are used the abbreviations are Snr and Jnr, respectively. The use of these social terms is governed by etiquette, but not enshrined in law. According to The Emily Post Institute, an authority on etiquette, the term Jr. can be correctly used only if a male child's first, middle, and last names are identical to his father's (current) names. When a male child has the same name as his grandfather, uncle or male cousin, but not his father, he can use the II suffix, which is pronounced, "the second".
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In the United Kingdom, the suffixes "Snr." and "Jnr." are rare, and not usually considered part of a person's name as such. Ordinal suffixes such as "III" are generally reserved for monarchs; however, the General Register Office has stated that, whereas it would normally reject a string of symbols or letters that "has no intrinsic sense of being a name" when registering a child, a suffix such as "III" would be accepted. Those who inherit a title of nobility do not use ordinal suffixes, but are distinguished from any ancestors with the same name by their position in the order of succession; for example Arthur Wellesley, 2nd Duke of Wellington is thus distinguished from his father, Arthur Wellesley, 1st Duke of Wellington.
Where I've met people in Ireland who are named after their father, they generally just use a variation of their name, e.g. Michael and his son Mick.
I've just realized that the example I gave above also applies to German racing driver Michael Schumacher and his racing driver son Mick Schumacher. I honestly wasn't thinking of them when I wrote it.