yeah, no idea how exactly it was sorted out (or if one of the names is an "official" first name), but if he calls himself Alexander on his social media profiles, I guess that's good to go with.
You would imagine that they have some way of establishing clarity for family gatherings etc. Alexander is the younger, so I would imagine the family habit would be to give him the alternative name rather than re-naming the one they had already been addressing as Jefferson for 2 years.
But how little imagination does one have to have to give your son the same name as your brother gave his son? (I guess there must be some family significance)
There are plenty of riders (or any other section of society) that are commonly known by their second given name (even excluding strange Danish naming customs): odd that commentators haven't been keen to prefer this one.