There are many constitutional monarchies in Europe. However, I never see the enthusiasm people show for them as they show for the British royal family and their affairs.
Why is that?
The reason for the great enthusiasm (or to be a bit more fair, interest) shown worldwide in the British monarchy compared with other monarchies is simple - worldwide reach and influence.
A huge number of countries have at least some connection with the British monarchy in the past - India, Canada, Australia, much of Africa and the Caribbean, and significant parts of the Far East. Even for countries that have long dispensed with any formal or political link with the UK, there will be people who have grown up knowing about the British monarchy, if only from their history lessons. Commonwealth countries will at least have known the Queen as "Head of the Commonwealth" and many will have been visited by her. Even in the US there is a connection built on a common language, constant references to the British monarchy in history (even if set up as "the enemy") and significant shared culture (not to mention through recent marriages).
Far fewer countries have a historical connections to Belgium, the Netherlands or Thailand.
The UK is also a larger (or at least richer) and more influential country than many other monarchies. More people are interested in the UK Monarchy than the Swedish Monarchy for the same reason that more people worldwide are more interested in the President of the US than the President of Turkey.
The King of Spain is only the King of Spain. The King of the Belgians is only the King of the Belgians. The same goes for all the other reigning European royal families - Denmark, Norway, Sweden, The Netherlands, Andorra, Lichtenstein, Monaco, and Luxembourg.
The King of the United Kingdom is also the King of Canada, Australia, Papua New Guinea, New Zealand, Jamaica, and nine other nations.
The British monarchy currently has 151 million subjects, and shares a language with 1.1 billion people. The next-largest European monarchy (the Spanish) has only 47 million subjects and shares a language with 548 million speakers.
Naturally, the British Royal family will have the greatest global interest, as it it more relevant to more people.
The Monaco royal family attracted significant enthusiasm in the mid-20th century, but that's only because its Princess was also a major Hollywood film star.
The other European countries are a bit more democratic. For example the UK has an upper house whose members are hand-picked, that does not happen in the other countries. Some might have a senate whose member are not elected in a direct manner as it happens in the Netherlands, but, still, they are elected.
An eventual change into a republic obviously would have a cascade effect on all the other institutions of the country. So the people currently in powers in the UK have a lot more to lose than the other European countries in case of a change and they invest a lot more in a media driven cult of personality to sustain their position. The difference in the amount of time dedicated by all the media in the Western world to the British royals in comparison to the others is enormous and the perceived popularity is just a consequence of their visibility in the media.
Furthermore through both tourism and foreign labour British people are very frequently in contact with foreigners. They see that so many foreigners know so much about their royals that they get from this fact an appearance of importance without thinking that this is just something induced by the media bombardment.
This is because England had its revolution a long time ago with the English Civil War, the aftermath of which which established the principle of the sovereignty of parliament. This was in the 17th C and hence the establishment of a constitutional monarchy. They have had a long time to get this to work well - as we saw today with the late monarchs funeral.
When the Russian Tsar Nicolas was faced with a similar revolution, he promised a similar constitutional settlement that was wildly welcomed by the then socialist and liberal factions but failed to deliver, thus prompting actual revolutions that abolished the Tsardom itself.
I expect something similar - but I could be wrong, as I haven't checked - happened with the other European monarchies - they didn't see the winds of change coming and thus the notion of a monarchy was buried. But then again - neither did the English - they had a civil war.