A naive understanding of family names (i.e. last names in English) might make you think that the number of unique family names in the world only decreases over time. Children get one of the two last names of their parents, and so no new family names are created and sometimes family names die out.
Of course, there's plenty of reasons the above isn't always true. Every existing family name had to start somewhere, and I know that process didn't end in the modern era. I have friends who have legally adopted entirely novel family names for personal reasons. Immigrants often change their family name, and the same original name may be changed into several variations. And I know from the example of Turkey that some groups of people began the modern era not using inherited family names and then switched to that practice.
I'm not expecting any historians have been keeping a careful count of the global number of family names, but I'm wondering if you can point to the modern-era events/trends that have likely resulted in the biggest change in the number of last names globally.
Am I right in guessing the number of family names has grown rather than shrunk?
Is the cause mostly immigration? Mostly specific last-name-generation events like the Turkish surname law? Or something else I haven't thought of?
Was the growth in the number of family names greatest early in the modern era? Or is it faster today?
Surnames nowadays function basically the same in every country—they're hereditary, a way to distinguish between people, and keep track of who might be related to who.
For a long time, hereditary surnames were only for nobility and were introduced to England by the Normans. In 9th century England, for example, a man might be known as 'John the smith' because his name is John and he's a blacksmith; but his son wouldn't then be known as Richard the smith. Probably he would be 'Richard son of John' or 'Richard the baker' or whatever his profession happened to be. Over time, though, they did begin to be passed down and became more recognisable as the surnames we know today: Smith, Johnson, Baker etc. This BBC website dedicated to family history has a few sections about the etymology of various surnames and how surnames came into use in England if you want to know more.
A common myth is that a lot of American surnames were corrupted by government officials who either didn't understand what immigrants were saying or couldn't be bothered to spell their names out. There's a very interesting Smithsonian article about this and Ellis Island in particular, but the relevant part is that most of these changes were actually made overseas when the immigrants were first setting off from their home countries—a clerk might misspell their name in the ship's manifest which would then get handed over to the American government and transcribed in their records as was.
The modern era as a whole contains the biggest 'boom' in surnames as various countries began to make their use mandatory—which I would say is the reason for the biggest change in the number of surnames, rather than immigration or people nowadays coming up with new surnames. Interestingly, there's no law in the U.K. that says you have to have a surname (though you would struggle without one), but because patronymics fell out of use when the Normans took over, pretty much everyone already had surnames by the time the modern era began (roughly 1700-1750 depending on who you ask). This is why Sweden, which made surnames mandatory in 1901, has so many 'Anderssons' and 'Johanssons'—people just used their patronymics as surnames.
Naturally, from then until now the number of surnames has only increased. The vast majority of countries use surnames now and the ones that don't (e.g Iceland) tend to have very small populations. Though, as you say, it's hard to get an exact number. Surnames go extinct all the time and it can be difficult to tell whether that's being counterbalanced by another country's naming practices. You would be incredibly hard-pressed, however, to find someone who thinks that there are fewer surnames floating about now than there were in the 1750s. And quite obviously they would be wrong.