Because the British Army was historically assembled by noblemen who raised their own forces, whereas the Royal Navy was established and funded by the Crown. When it came time to invent an Air Force, all of this stuff had long since become an organ of the democratic British government, so they just called it Royal to sound cool.
The commonwealth countries inherited the nomenclature.
Basically the components of the English and Scottish Armies were raised by liege lords of the respective monarchs, rather than the monarch themself. This all started changing with the New Model Army, and then the reign of Charles II, but the non-royal association stuck. The Navies (well, the Scottish navy was negligible) were financed a lot more centrally. I’m not sure why the RAF took the Royal name, and presumably the former Dominions just followed the British convention.
If you look you will see that some regiments of the British and Commonwealth armies are themselves “Royal.” The Sovereign is the nominal colonel of those regiments, including the Regiment of Marines and old line outfits like the 42d Regiment of Foot, the Black Watch, the Royal Highland Regiment. Apparently it was once a big deal for the King/Queen to express a wish to be the regimental commander. The give away in the dress uniform is blue facings – colors and cuffs – only the royal regiments get those.
Doesn’t the creation of the New Model Army by the Parliamentary side of the Civil War also have something to do with the historical reason? The first modern British professional army wasn’t Royal at all, quite the opposite in fact.
I can’t speak for the other Commonwealth countries, but the ‘Australian Army’, largely includes the royal label.
As in, all infantry in the Australian army is contained within the Royal Australian Infantry Corps. Other arms are similarly named as in the Royal Australian Armoured Corps and the Royal Australian Artillery
So while the Army as such is not labelled as being royal, the major divisions within it have the ‘royal’ label.
As mentioned by Spavined Gelding, the Queen (of England ) is the Colonel-in-Chief of the RAIC.
Technically Canada no longer has a Royal Canadian Navy, Royal Canadian Air Force, or Canadian Army. Since the '60s it’s been; Candian Forces Maritime Command, Canadian Forces Air Command, and Canadian Forces Land Force Command.
While not all of the British Army is part of a Royal something, a good deal of it is. For example, the Royal Corps of Engineers, of Signals, and of Transport, the Royal Tank Regiment, The Royal Artillery Regiment, Royal Regiment of Fusiliers, Royal Scots Regiment, Royal Irish Regiment, Princess of Wales’s Royal Regiment, Royal Hampshire Regiment, Queen’s Royal Surrey Regiment, Royal Highlanders Regiment, Royal Anglian Regiment, King’s Own Royal Regiment, etc.
In looking up something completely different about Elizxsbeth the late Queen Mother a couple weeks ago, I noticed that she had been honorary Colonel-in-Chief of the Black Watch from 1937 until her death – 65 years!
This is somewhat misleading - but the time a permanent English (it didn’t become British until the Union with Scotland in 1707) Army was established in the mid-17th century, the days of noblemen raising feudal levies from their own estates were long gone.
After the Wars of the Roses, the Tudor monarchs cracked down hard on “over-mighty subjects” raising their own private armies. The English Army of the 16th century was basically a national militia, organised by counties and towns and raised by royal officials (and definitely not by local bigwigs who might get ideas above their station).
The reason the army of Charles II and his successors didn’t become the Royal Army was that his political opponents wouldn’t stand for it. The fear was that given a Royal Army under his direct control, a monarch could use it to suppress public liberties and restore absolutism. (Many Whigs opposed a standing army of any sort for this reason - it comes into Gulliver’s Travels for instance).
The same fears did not attach to the Royal Navy - it’s hard to impose martial law with ships.