It differs per kingdom. I believe Belgium only has to do army. The heir princess graduated the royal military school a couple of years ago, and apparently has a talent for it. Lots of praise for her.
And here in the Netherlands the monarch is not head of the army/navy/air force, so I guess it is not required. Our heir princess is at uni (between getting threatened by the underworld to get kidnapped), studying Politics, Psychology, Law & Economics. Never heard anything about her going to a military academy.
We now have a king after almost a decade of queens, in their time it probably wasn't even an option to do a military training. I have read that Queen Wilhelmina (during WWII) regretted not being able to.
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