Yeah, it's a bit weird...
On some forms in the US, it's listed as a suffix. Some agencies enforce it, others do not (SSA does not, for example). And colloquially, when"Senior" dies, the son often stops using "Junior" (though that's not a legal change).
UK, Can, and AUS are similar, but not exactly the same as the US. And NSW in AUS complicates matters by outright prohibiting roman numberals are Jnr/Snr suffixes. Mexico has more restrictions due to the mothers-last-name, fathers-last-name naming convention.
Personally, I dig the Nordic and Arabic "x, son of y, son of z, son of..." patterns.
The UK doesn't really use those kind of suffix at all unless you're the king, so I'd imagine it would be quite hard with forms.