Is this a normal way to install a tire?
It's hard to tell exactly what was done and why – it would help to know the original and new tire sizes. But what it looks like is that the shop added inline barrel adjusters to your brake cables. That's consistent with your description as well. They probably added them because the release levers on the brakes didn't give enough clearance to remove/install the wheels with the new tires without having to undo the cable clamp bolts. This is to make it easy for you to repair flats and the like.

I would have expected the shop to contact you and tell you that the new tires were too wide to work with your current brakes and offer you options for the solution. One solution would have been to use a narrower tire, another is adding the barrel adjusters.