Rarity is part of it. Other properties are relevant as well, though. It’s shiny (or, as the professionals would say, lustrous), always important in picking valuable stuff. It’s both ductile and malleable, so it’s easy to make into jewelry or coinage. It’s also extremely nonreactive, which means it doesn’t corrode or tarnish, both appealing qualities for a ‘precious metal’.
Incidentally, the crown of Queen Elizabeth (the Queen Mother, not the current monarch), is made of platinum.
Alumin(i)um was once worth the same as silver, though this is less a function of its “precious” nature than the fact that it was hard to work with so nobody refined it.
Three years ago, for a little while, the spot Gold price was higher than the spot Platinum price. But usually, Platinum is a little higher. Today, gold is about 0.89 of the price pf platinum. Their prices rise and fall according to different market forces.
It is just pedantry to claim there is no queen of England when, in fact, as part of the United Kingdom, England DOES have a queen - even if no individual since 1707 has had the title “Queen of England”.
Sometimes we refer to her as “Queen of England” simply because it’s shorter than saying “Queen of the United Kingdom of Great Britain and Northern Ireland, and her realms overseas.”
She’ll be the longest reigning monarch in the British Isles then. Elsewhere, there have been longer reigns.
Sobhuza II of Swaziland had the longest verified reign: just short of 83 years (1899-1982), beginning when he was four months old. Louis XIV of France (72 years, 110 days) had the longest reign among major European monarchs, although several rulers of minor German states had longer reigns.
The present Queen isn’t even the longest-reigning current monarch: that honor belongs to Bhumibol_Adulyadej of Thailand, who became king in 1946 (six years before Elizabeth II).
Then there is the curious case of Michael of Romania. He first became king of Romania in 1927, at the age of six; almost 87 years later, he’s still alive and kicking. (Of course, what with Romanian dynastic politics and history what they’ve been, he’s only actually been king for ten of those 87 years, and exercised substantial political power for maybe nine months.)