The proper repair for a simple tread puncture is Plug + Patch, from the inside of the tire. did the shop do that, or just a plug? I ask because you the owner cannot verify a patch repair on the inside unless you actually observed the repair in progress.
The plug is to fill in the missing rubber where the puncture took place and is inserted from the inside of the tire, and has a small retainer lip on it, and the patch insures that the tire is fully sealed and safe to drive on for the remaining life of the tire. Both are glued in the repair process to form a permanent vulcanized rubber seal.
A plug is fine as a temp repair to get you to a tire shop (I carry a set of plugs and tools to patch a simple puncture with the tire still on the vehicle, as well as 12v powered air inflator). But a permanent repair requires filling the hole from the reverse side with a plug and then bonding a repair patch over the inside surface where the plug is located. Tire shops will also grind the inside rubber surface a bit to help get it clean for permanent gluing.
If replacing a single tire, it must be the same brand, model, and size of tire or your AWD will go nuts, make noisy driving, and ultimately may fail under the stresses of mismatched tires. It also must be close to the rest of the tires in diameter, so generally you would only replace a single tire if it was still with good tread life matching the rest of the tires (~70% remaining tread is fine as is. If there is less tread on the other 3 tires than this, the best practice is to replace the pair of tires on the same axle, and insure they are always on the same axle when rotating tires periodically.