Its worse than just the swans - the monarch owns almost all the land in the UK too. The legal concept of "Allodial title" [1] means that, even if you own the freehold of some land, it's actually a freehold in the estate of the crown [2] - and the monarch is actually the ultimate owner. The exceptions to this are a few areas of Scotland.
Actually, this doesn't apply to the whole of Scotland since "The Abolition of Feudal Tenure etc. (Scotland) Act 2000", and it doesn't apply to NI either. The crown ownership only applies to England & Wales.
If I remember right, the current Princes of Liechtenstein basically bought the current country under a similar concept.
The original Liechtenstein was a castle in Austria. The family shopped around for some land that was held directly from the Holy Roman Emperor, rather than through some intermediate feudal lord, and bought it. That gave them statehood within the Holy Roman Empire, and raised the head of the house to the status of Prince.
They didn't even visit the current Liechtenstein for many years -- it was basically a flag of convenience.
You’re phrasing this like it actually has some bearing on how land is owned in the U.K. This is just a quirk of tradition - for all intents and purposes the freeholder is the ultimate owner of the land.
The Crown Estate is one of the largest landowners in Britain, with peers including the Church and the Ministry of Defence, but does not own the majority of the UK. Curiously, a lot of the coastal seabed in UK territorial waters is part of the Crown Estate, which has thus profited massively from offshore wind farms.
The Crown 'owns' the Crown estate as a corporation sole [0], a legal entity occupied by a single person, the Monarch. When the Queen died, the Crown persisted as a legal entity but the Monarch changed. King Charles owns several properties that are not part of the Crown estate, e.g. Balmoral and Sandringham. These were purchased by members of the Royal family from other members of the aristocracy in the 19th Century.
[1] https://en.wikipedia.org/wiki/Allodial_title
[2] from memory - the actual phrase might be slightly different