I don't have one, nor do I really want one. But I have handled and shot a few rounds. Seems to work well, and ergonomy of the stock was really working for me, could not really percieve any difference in 300WM with and without a (decently sized) suppressor mounted. My own observation of their popularity in Scandinavia/northern Europe:
Of importance (real and percieved)
* As have been said, in places where there are limits on how many guns you can own, it can be so that a switch-barrel gun only counts as one. This adds to popularity.
* Due to the construction, the overall length of the assembled rifle can be a couple of inches shorter without sacrificing barrel length. Popular in general, especially so in this time of suppressors, and even more so for the forms of driven hunts, where a dog handler follow his/her dog through the forests in pursuit of the quarry. It will not protrude as far above the shoulder as a 'regular' rifle, which makes it handy when following through thick brush. Also makes for a short handy rifle in general.
* Barrel-mounted optics with better chances of keeping the same point of impact between barrel changes.
* Fashion. Simple as that. Blasers are the new black. To some extent there is a silent bragging factor not to be ignored. It shows that you have a bit of money to spend. I believe that the social aspects can be a bit different, as it is quite common to end up on a hunt through a friend (or a friend of a friend), where there are 15 other people that haven't met you (or necessarily each other) before. So it is not always you and your closest few buddies that go hunt for a day.
* The safety/cocking. But I still think that muzzle control is more important.
* Can be had in left-hand configuration (but it will cost you extra)
Of percieved (but not real) importance for choosing a Blaser (R8 or R93)
* Speed of the follow up shot. I really do not think that the extra 0.25 seconds (or whatever) compared to a traditional repeater are a factor in reality. And for those driven hunts of boar and other non-dangerous game, people can (and sometimes do) get a semi-auto (where legal).
Of less importance to the popularity, but still an observation on what makes Blasers good rifles:
* Barrel-steel quality is supposedly very good, with excellent hardening/tempering for optimal surface hardness without being too brittle.
* Out-of-the-box precision. Not really surprising, as modern manufacturing makes this achiveable for far cheaper guns. Like Tikka T3, Sauer 101 etc.
* Compact storage. But I don't think that most owners actually do utilize the take-down feature all that much.
Probably a few more things to mention for the reasons for getting a Blaser, but... for some reason I've never really wanted one. No doubt that it is a quality product, with some excellent features. Just feels that they lack a bit of soul, I think. Sort of an engineers solution to an artists problem.