This was quite common, not just for boys, and also along many strata of society.

There are two things to consider here:

a) Living at the royal court was considered a rather unhealthy environment for a child:

Lots of people who could get ill, questionable hygienic conditions, bad air, perhaps immoral going-ons etc. at a royal court.

So royal children usually received their own (smaller) court somewhere in the (supposedly much healthier) countryside. When Edward VI. of England was a baby, his court included a lady governess, treasurer, chaplain and a troop of his own musicians, along with a wet nurse, several nannies and even women, whose only job it was to rock his crib. When he got older private tutors joined his court to educate him. Until the age of seven boys were raised among women with a lady governess in charge. After that they started their formal education and there would usually be a man in charge of his court at that point.

Keeping the child's court close to the adult royal court to ensure familial visits was not possible. Like the adult court, a royal prince's or princesse's court had to move to a different place regularly, while the old place was cleaned and aired out. And also because the amount of people at a court was such a huge drain on local resources that they needed to go bother someone else with their presence for a while.

b) Showing off your status:

It was very important for royals and the nobility to project their power and importance to everyone around them. That meant dressing lavishly, keeping many servants, always being accompanied by your retainers etc. - in essence, as large a court as possibly to show your status. (Even a duchess at this time could hardly go anywhere without a preceding herald and a train of servants to be considered "presentable". These people were never alone!)

A crown prince, who was naturally a very important person, needed a lavish court of his own. Prince Arthur, Henry VII's eldest son, was raised at Ludlow in Wales at his own court for example, while his younger siblings were mostly raised together at Eltham in a sort of joint, more humble court, where they were regularly visited by their mother, Elizabeth of York (the elder sister of the Prince of Wales you saw being sent away in The White Queen)

Now obviously, when your child is living in pomp and circumstance in the healthy countryside, away from your own glamorous, but unfortunately unhealthy court, and you can't visit them all the time, someone needs to be in charge of teh considerable logistics necessary to keep your child's court from running smoothly.

And that's where the guardian comes in. For girls all throughout their childhood and for boys until age seven a lady governess ran their court, for boy a male guardian came in later.

As far as I know this person would only be called a guardian if the child's father was dead though.

If you were the son or daughter of, say an earl, and your father died during your minority, guardianship of you fell to the crown, who would immediately sell it off to other noblemen who then had to take care of you during your minority but also got to enjoy the income from your lands and property and decide who you had to marry (usually their son or daughter of course!).

For the minor nobility however, it was also very sought after to send their children off to "serve" at the court of a higher ranking noble. This was especially important for younger children, who would not inherit a title, in order to make connections and essentially start their courtly career, since that was their only way of getting ahead in life, either by later receiving a good job at court (there were plenty of those reserved for nobility) and/or by finding a good marriage prospect. Usually both for both genders.

A noble girl for example would start out as a maid of honour to a baroness, duchess or princess/queen at age 12 and get paid for it. That was the "lowest" job available, and entailed menial work like helping her mistress dress, washing her feet and such. Later in her life, she might become lady governess to a young prince or princess herself, a job with a lot of prestige (and much better pay).

Equally, a boy would start out as a page to a baron, duke or king, equally get paid and later rise to more important positions at court. As long as they were children, their "employers" acted as their guardians in essence.

So sending your children off to live with other people was a very common practice, though for different reasons.