I don't let Facebook show my personal info.
You mean you haven't fully given in to the might of Facebook?! Succomb, it's fun :D
I don't think the British Army is "Royal" either, being the red-headed stepchild of their force structure for most of it's existence... There are some Royal units, but as a whole... meh?
For much of history, The Army never exsisted as an entity, but was a colection of Indaviduly rasied regiments, under a commen leader. As time went by thing became much more integrated but tradition remains the same.
Pretty much, we still don't really have a standing Army - it has to be voted for each year as a separate law as an army loyal to the Crown alone was seen as a tool of oppression (a Royal Navy was a seen as a tool of oppressing other people and hence a Good Thing)
Many of the Regiments and Corps that make up the Army are Royal, like the Royal Artillery and Royal Regiment of Fusiliers, but many are not, such as the Army Air Corps and The Rifles
The Army is still considered a collection of Regiments and Corps rather than a single unified entity in the way that the Royal Navy and Royal Air Force are - a soldier's loyalty to their Regiment is supposed to be far more enduring than that of a sailor to their ship I believe (at least historically)