Actually most people have no idea about the AR designation and besides "assault rifle" next most commonly think that its origins came from Armalite Rifle, which again is incorrect, but the most accepted claim especially with the advent, popularity and trademark of the AR15 designation. Very few people know the true origin. During the development of this platform it was originally a "heavy" or "battle rifle" class in a 7.62/.308 caliber weapon that was indeed a select fire variant. Since it was a "heavy" or "battle" class of rifle and not a lighter caliber fitting the "assault rifle" category there was absolutely no "Assault Rifle" meaning behind the original "AR" designation. It would have been a BR10 instead if it was referencing the class of rifle, but it was not.
Why the "AR" moniker confusion? In 1959 when Armalite / Fairchild split Colt got the AR15 moniker from Fairchild and Armalite went to George Sullivan. Colt trademarked the AR15 moniker and eventually several other terms. However Colt went on to develop a civilian semi auto only version of the M16 and designated it the AR15 since they already held that name designation. Definitely not a true reference to "Armalite Rifle" as they did not own that name and did not produce that rifle. It also did not stand for "assault rifle" as it was a semi-auto only version. Colt holds the AR15 Trademark and attempted to own the "M4" designation also, but lost that one in court as it was a military designation and quite frankly they did it too long after others were using it also. Technically I don't think Colt holds any more "patents" on the platform but they do have the AR15 name trademarked as well as several other trademarked names. They did at one point hold the contract for the Military Specification "Mil-Spec" rifle award, but now others are producing under Government contract also.
So now we understand that "AR" originally went back to Armalite pre Colt and pre Fairchild. So it must have meant "Armalite Rifle"? Well not quite. Armalite was developing an automatic rifle in the 7.62 / .308 or battle rifle category. The original automatic rifle design went through 9 different variations before the last or the 10 version was finalized. So it was simply the 10th version of the "automatic rifle" design, or "Automatic Rifle version 10" and later shortened to simply "AR10", so while it does not mean "assault rifle" the original "AR" term did reference a select fire rifle.