Does the British monarch really own all of the swans in the UK? If so why? 
In like the 12th century, the British crown claimed ownership of any unmarked, wild, mute swans. It kept that clain until until the 15th century, when it sold a lot of its swan rights to various guilds and other people, but the crown still claims ownership of all the swans on parts of the River Thames, and every year, young swans on the Thames are caught, marked and released, although nowadays, they do it for census and tracking purposes.
Why did the Kings of England do this? Swans are tasty.
The Queen also still owns all whales, dolphins, porpoises and sturgeons up to three miles out from land.
No, they only claim rights to the swans on certain stretches of the Thames.
From the Royal Website
This dates back to the days when swans were a prized table bird. The Crown reserved some for its own use.
More here.
I hear swan is really greasy.
That’s no doubt why it was considered “fit for a king” back in the 12th century. Good eats! 
If I were King of England back then, I’d definitely want some of this Swan(n): http://www.forbiddenplanet.co.uk/images/P/PO18.jpg
I thought those belonged to the Prince of Wales.