There’s actually a few reasons for this.
First, shotguns are very easy to handle. In spite of their destructive capability, they’re relatively low power by firearm standards. In fact, some non-standard loads, (Dragon’s Breath, for example) don’t generate enough power to cycle the bolt when fired. This means they deliver comparatively little recoil, and make follow up shots (even on a pump action) remarkably easy for an inexperienced user.
This does mean, when using some non-standard loads, your characters will need to manually cycle the action after each shot, even on a semi-auto shotgun.
The second major thing is that shotguns are extremely versatile. There are a lot of different possible shotgun shell loadouts available, ranging from the conventional buckshot and slugs to the exotic rounds like flares, flechettes, piranha (think a shotgun shell loaded with thumb tacks) and dragon’s breath rounds. There are even bolo shells, with two heavy balls mounted to a wire, designed to cut through anything that gets between them. Though, bolo rounds aren’t entirely reliable. We’ve mentioned them before, but there are also the FRAG12 rounds, which will convert a shotgun into a grenade launcher.
This is before you consider hand loaded rounds people assemble in their spare time. These range from cut shells (which convert a buckshot shell to discharge the shot inside the victim) to magnet clusters or even unspent airgun C02 cartridges. (I don’t recommend attempting the C02 cartridges on a commercially produced shotgun.)
When you’re bringing firearms into hunting monsters, you’re often looking for specific weaknesses. For example, a vampire may be immune to buckshot, but what about a shotgun shell filled with toothpicks? It’s going to splinter and be a mess, but it will still drive those wooden shards into the monster, and have a pretty decent chance of piercing the heart.
That said, wooden slugs don’t really work. Most are too light to fly properly. (This is also an issue with silver bullets.) Though, this might be fixable by someone with enough time and an actual application in mind. Some kind of wood core sabot might be another option.
Third, contrary to popular perception (and most video games), shotguns are useful at medium ranges. In theory shotguns remain functional at 100 yards, though 50 yards is a better effective range estimate. This means, yes, you can put a shot shell into a nine foot tall snarling deathbeast at 150 feet.
Finally, shotguns are easily available to most characters. Your character isn’t going to be breaking out a full-auto AA-12, but they can probably get their hands on a Remington 870, Ithaca Stakeout, Mossberg 590 or something similar.
They’re far easier to obtain, and a Winchester 1300 loaded with rock salt shells is
far less likely to draw unwanted attention, the way most specialized monster hunting gear, or high power weapons would.
They’re not a perfect solution to every problem, and obviously just won’t work for some characters, but shotguns are probably one of the most versatile tools an urban fantasy monster hunter can get their hands on.