In the Netherlands, Belgium, Sweden, England and a whole pile of other countries the royal families are pretty well entrenched. There is a whole media circus around them and there are still special laws on the books giving the royal people rights that ordinary people do not have.
The most heard argument against abolishing these institutions is that 'you need a head of state' and 'a president costs money too'. I'm not really sure what either has to do with creating and maintaining a special class of people, and I can't imagine a presidency to be as costly and as messy as what we have today.
Besides that, 'our' royal family isn't even Dutch, but hardly anybody seems to care, as long as the fairy tale is allowed to continue it's all good.