Joachimsen would refer to "son of Joachim". This is similar to Henriksen, which would refer to "son of Henrik".
I know (Iím Danish, after all
) but you canít just use a last name that isnít legally yours. Especially not one that may be a protected name (although I donít know if Joachimsen is that).
I completely get your argument, but then, there are so many 'royals' who still use their former titles. For instance, Pavlos and Marie-Chantal and their kids, or Anne-Marie and Constantine, French royals, German royals. However, I don't quite know whether it's irrelevant in their case since they are royals of defunct kingdoms. Having said that, I think you could argue that even if they have no 'kingdom' the title still belongs to their State and they still shouldn't use it. I know for a fact, being half-Greek, that the Greek royals aren't allowed to use the title in official documents, e.g. passports. What they are allowed to use is the surname 'de Grecia'. But that still doesn't prevent them from calling themselves Crown Prince or Crown Princess, King or Queen, everywhere else!