I think in earlier times this habit served to help the subjects to pronounce the new spouses' names, esp when the names would be very difficult for people to say. But it also solidified the fact that this new princess/queen (and for most parts it was women) was now a "native" and not a foreigner. (Eg the unpronounceable name of Dagmar was switched to Marie (using the last syllable only) in Russia).Didn't work very well, though.
I didn't know Claus' other first names had been changed into dutch as well, was that the case with Bernhard as well? I know that Prince Hendrick was likely a born Prince Heinrich.
Prince Hendrik, the spouse of Queen Wilhelmina, was indeed born as a prinz Heinrich:
Heinrich Wladimir Albrecht Ernst Herzog zu Mecklenburg = Hendrik Wladimir Albrecht Ernst
Hendrik is the general Dutch version of Heinrich.
Prince Bernhard, the spouse of Queen Juliana, had only part of his names changed. His first name stayed the same. As far as I know at least 2 versions of the name exist and or are used in Dutch: Bernard and Bernhard. But I guess the first version of Bernard is the most "Dutch" one.
Bernhard Friedrich Eberhard Leopold Julius Kurt Carl Gottfried Peter Graf von Biesterfeld = Bernhard Leopold Frederik Everhard Julius Coert Karel Godfried Pieter, Prins der Nederlanden, Prins van Lippe-Biesterfeld
It also seems his names were also a bit mixed up
-------------------------------------------------------------------------------------------
Philippe of Belgium was previously in the Netherlands predominantly known under the Dutch (or Flemish) version of his name Filip(s).