I have written about the topic of these kinds of names before here.

Those suffixes are actually an early form of last names called a "byname" or an "epithet". Our modern version of fixed surnames inherited down the male line, in fact, is a result of William the Conqueror's invasion of England. There weren't fixed surnames anywhere in Europe until ~1066, at which point they began to come into use in England and France.

William the Conqueror in his time was known much the same way as Queen Elizabeth is known today. William was the son of Robert I, Duke of Normandy, and then William himself became the Duke of Normandy. He would have been known to most people contemporaneously as "William of Normandy" or simply "William" or perhaps "Duke William" if you lived in Normandy or "Duke William of Normandy" if you didn't. And then when he was crowned King of England, he would have been known as "William", "King William", or "[King] William of England" just as Queen Elizabeth is known nowadays as Elizabeth, Queen Elizabeth, or occasionally as Queen Elizabeth II of the United Kingdom and Great Britain.

After his death, as there came other King Williams of England, just calling him "William" would not always suffice, so other epithets came into popular use: "William the Conqueror" if you liked him, "William the Bastard" if you didn't, or simply "William the First".

People probably didn't call William much of anything but William to his face, but those nicknames "conqueror" and "bastard" were promoted early on. The official chronicler of the pro-William side of the invasion was a guy named William of Poitiers, whose writing on the subject is known as Gesta Guillelmi Ducis Normannorum et Regis Anglorum ("The Deeds of William, Duke of the Normans and the King of the English") and in it, he refers to William as simply "William".

But by the time William had died, a hymn and an epitaph written upon the occasion referred to him as "William the Conqueror".

Shortly after his death, however, the Anglo-Saxon Chronicle referred to him as "William the bastard". And then not long after that, 12th Century historian William of Malmesbury referred to William as simply "William the First".

By the 1300s, writers were routinely referring to him as William the Conqueror. And it's the name he's most popularly known by, though it has always remained interchangeable with William I, at least since his son William II came to power.

The same kind of thing happened with most of those people you see in history books with those kinds of bynames, though the story of when and how it came to be their byname will be a little different with each historical person. Typically, the people remembered by history of that time were royalty, nobility, or high-ranking clergy, and they would have been known to their contemporaries by their title, as explained above ("William of Normandy", "Baldwin of Flanders", etc.).

For the rest of the population, for a long time, they were just referred to by a first name, but by the time of William the Conqueror, they, too, were often using impermanent bynames such as "John the Smith". If you read through this previous post of mine, it explains how this system worked. Occasionally, someone who wasn't necessarily royalty would become famous outside of their local community, and they would get a new byname different than the one they might have been using locally. For instance, the aforementioned William of Malmesbury was locally known in Malmesbury as something else, now lost to time, but probably most often simply known as "Brother William", since he was a monk at Malmesbury Abbey. As his name spread throughout England due to his writings, people outside of Malmesbury who needed to differentiate him from the thousand other famous Williams called him "William of Malmesbury".

This kind of system lasted in Europe for a long time. For instance, by the time Leonardo da Vinci was born in 1452, his birthplace of Florence was using a system of last names called patronymics (still in use in Iceland today), so his birth name was "Lionardo di ser Piero" ("Leonardo [son] of Sir Piero"). But since he became so famous, and nobody would have any frame of reference about his father "ser Piero" outside of his local community, Leonardo was known around Europe as "Leonardo da Vinci", literally "Leonardo of Vinci" since Vinci was the name of his hometown.

So, no, these names weren't really "taglines". They were simply the system of last names called "bynames" and this is how they worked. Most often, you were known as "George, John the Smith's son" until you reached adulthood. If you stayed in town and became a blacksmith just like your father, you'd probably keep that same byname and be known as "George the Smith". But if you moved away, or you took up a different occupation, then you could be known as something else entirely: "George of London" or "George the Cooper" or "George the Brown" or "George at the Wood" or any number of other fitting alternatives.

This is how it worked for nobility and royalty, too, as is the case of William the Conqueror. The difference was, their byname was usually simply their noble title, but since these people were often political figures, they might have informal bynames adopted by their supporters and detractors, one of which may have won out over the others in the history books, most often because their name and title was shared by their ancestors or offspring. Thus, instead of simply "William of Normandy" or "King William of England", he's more often known in history books as "William the First" or "William the Conqueror" to more readily separate him from later King Williams of England, who had their own, often posthumous, bynames. His son William is most often known as William II or William Rufus, which is another way of saying William the Red. Other royalty also had informal bynames that never became as ubiquitous. For example, Henry I was also known as "Henry Beauclerc" to writers after his death, but nowadays, history books will usually just call him "Henry I".

The interesting coda to all this is that, while the rest of the English population moved to fixed surnames during the 1100s-1300s as each local landlord or noble began formalizing them among the peasants for tax, inheritance, and transactional purposes, the English royal family maintained the older byname system right up until World War I. It wasn't until then that George V adopted "Windsor" as not only the name of the royal house, but the family's actual surname as well.

There are sources at the bottom of my previous post here, which includes one that talks about the Norman Invasion's effect on naming conventions.

Side question: Did people call him William the Bastard to his face?