Moving blankets are so thickly insulated so they have more uses than just covering and protecting your furniture and many people have come to realize this. If you do have other uses for them then this might change your mind from deciding to rent them, to actually buy them. You can put them to use when you are using them for moving and then when you need to move again (god forbid) then you have moving blankets ready to go without worrying about going through the hassle of renting them all over again and dealing with deposits, credit checks, etc.
To answer to the question of whether they can be used for soundproofing a room, absolutely, as long as you have the premium grade quality moving blankets, usually made from cotton, then you should have no problem using them to soundproof a room or studio.
Many people have used thick shag carpet or other similiar items in the past, so the use of thick premium moving blankets to do the same thing shouldn't be unexpected. Many people use these types of products instead of purchasing high-priced acoustic treatments because it saves money, and makes use of something that you already had lying around. In some cases, people buy the moving blankets specifically for the soundproofing and don't ever use them to move!
When the sounds leave the speakers, the closer the blanket is to the speaker the less surface there is to reflect the sounds and noises coming from the speaker before it gets absored by the moving blanket. This gives you the ability to control how much of your room is covered, and how close they are to the actual speakers to determine exactly how much soundproofing you want. Some may want to place the speakers right next to the moving blankets, and have them double-covered, while others might not need that level of soundproofing and might just use one regular insulted moving cotton blanket, and place it fairly far away. Having this type of control is great.