Yes, that kind of last name is called a "byname" or an "epithet".
People have been using that kind of last name for thousands of years. Most of the non-Roman last names you see in the Bible, for example, are of this variety, including John the Baptist and Mary Magdalene (who is also known as Mary of Magdelena).
By the Middle Ages in Europe, these type of bynames had become a little bit more formal. Since a lord might have five different "John"s living on his manor, these bynames helped avoid confusion when it came to tax collecting and inheritence and other such matters. So, a byname was usually attached to a peasant around the time they became an adult/got married/started their own farm, and that name would be attached to that person their entire life. So even if "John at the Wood" later moved to a new farm by the creek, he'd still be known as "John at the Wood" for the rest of his life, because that's what the tax collector knew him as.
Fixed family surnames didn't arise until the Normans invaded England in 1066. There is some evidence that fixed surnames were in use shortly before that in France and Ireland, but not much longer--maybe a generation or two.
In much of the rest of Europe, the populations didn't go straight down the "fixed surnames" road. Instead, some adopted a "patronymic" naming system, such as in the Netherlands, Scandinavia, parts of northern Germany, Russia, Italy, and elsewhere. One by one, these populations adopted the modern "fixed surname" system, though that process played out over hundreds of years, and in Iceland, they still use the old patronymic system to this day.
How this system worked is that your father's first name became your last name. Your name is John and your dad's name is Peter? Then your last name is Peterson, and your kids' last name will be Johnson.
The problem with this system was that it led to a lot of people having the same first and last name as someone else who lived nearby, so these societies still kept using bynames as well. A full Dutch name up until the 1700s was something like "John Peterson the Smith" (or "Jan Pietersen de Smid"). In day to day conversation, people would usually just call you "Jan Pietersen", but the tax collector would insist on using your full name because three other "Jan Pietersen"s lived in the same town.
There were three kinds of bynames: geographical, occupational, or physical, e.g. "John by the Hill", "John the Tanner", or "John the Red".
Again, this byname was given to you around the time you got married or moved out on your own. Until that point, you would be using your father's. You'd be known as "Peter, son of John on the Hill" or else "Peter on the Hill".
Once you established your own home and family, your new byname was settled upon by your community. It wasn't something you chose yourself.
And you were stuck with it, unless you became very famous and got to be known as "the Great" or something, or else if you immigrated somewhere else. If "John on the Hill" who lives near Paris moved to London, he'd be "John of Paris" by the time he got there. It was possible for a peasant to end up with a byname such as "the Wise" or "the Strong" but it was relatively rare. Most often, peasants ended up with a more conventional geographical or occupational byname.
I have written on this topic before and my previous answer may be of some use as well.
And here is a book of transcriptions of documents relating to the King of Scotland in the 1100s and 1200s. Perusing the document, you can see that in the earlier entries, many of the last names still in use were not yet fixed surnames, but bynames. But by the end of the 1200s, at the end of that book, most people in the British isles had made the switch to the "fixed surname" system.
Sources:
The History of the Norman Conquest of England
English Surnames: Essays on Family Nomenclature
The World's Great Classics: The History of Europe In the Middle Ages
I'm not sure if this is exactly what you're looking for, but in Ireland it was/is very common to identify someone by a nickname that is long the same lines as those in your question.
This is a practice that starts very early - though I am hesitant to put a date on it we do have records in old Irish with warriors or chieftains having nicknames - and the most recent article I've read about it was published in 2010: Walsh, Colm. "Nicknames: A Directory of Occupations, Geographies, Prejudices and Habits." History Ireland 18, no. 3 (2010): 26-28
Especially in smaller communities where you would have an overlap of names in people that you knew, people were often given nicknames so that you could distinguish one Patrick from the other, say. Often, these nicknames were determined by physical features, or personal attributes. Colours were particularly common, usually the colour of the person's hair: Black Thomas had dark hair while Red Thomas had red hair. These are the ones that come up the most often, in my experience, but you will get things like Strong Tom or Bold Nora.
My personal favourite is the surname name Kennedy, which comes from the old Irish ceann éidigh meaning 'ugly head.'
Titles like the ones you describe did occur as well, usually in reference to someone's habit or profession. John the Doctor, Colm the Master, those sorts of things.
Off the top of my head the only article I can think of that discusses things more in depth is:
Breen, Richard. "Naming Practices in Western Ireland." Man, New Series, 17, no. 4 (1982): 701-13
However if I can scrounge up any more, I certainly will share.
Viking nicknames are also common, both in sagas in history generally. It seems that there's no one reason why a nickname sticks, it seems, but they follow the predictable pattern of describing some aspect of that person - and they can be inherited. They vary considerably, from the flattering to the astonishingly rude. They can refer to where you're from, what you look like and what kind of a person you are. They can be given to anybody, from any walk of life.
They include the obvious royalty in Scandinavia. Non-royals in the Iceland Sagas include Icelanders Egil Skallagrimsson (Egil son of Grimur the Bald), and Kveld-Ulfur (Night Wolf, in this instance his name was Ulf, meaning wolf, with the Kveld added. He was a pretty bad tempered guy).
Rude nicknames like Strað-Bjarni (Butt-fuck-Bjarni) are also seen, as well as Ásný Knarrarbringa (Ásný Big (ship) Chested) and some very nice nicknames like Helga Væna (the fine). Peple could also change their nicknames, although not always for the better.
Here are two theses: [one] (http://skemman.is/stream/get/1946/12799/31123/1/Old_Norse_Nicknames.pdf) and two. They go into a lot more detail about why people where given nicknames, and more importantly the second has a long list of amusing nicknames given to people.