If you are really itchy, it could be you are allergic to one of the fibers in the garment- you should see your surgeon to rule that out.
After liposuction, a compression garment can help reduce swelling and help to improve results. The garments used for medical problems are specially designed. If you look carefully at stretchy fabrics, you will notice that some only stretch in one direction (biased), whereas others will stretch in both directions. Special support garments are sewn so that the bias angle (the direction of stretch) is purpose specific. Some have several layers in one panel but not others. Your surgeon may not have explained what the purpose of your specific garment is, so just ask!
Another important feature is where the seams are, and whether or not they are padded or taped. If you had subdermal or subcutaneous liposuction done, then the garment should not have any thing poking into your skin, since that could leave a slight indentation when healed. Better medical compression garments are designed not to do that.
I have recommended that my patients use a tight cotton undershirt underneath the garment ( that is long enough to cover your skin and tight enough to not cause any wrinkles in the fabric that is against your skin). You should launder these undershirts daily, since about 90% of your dead skin cells and body oils will be absorbed by your undershirt, and it is best to change it daily (this is from NASA basic research into clothing back in the '60's). Bleaches (usually chlorine based) always damage the stretch fibers in your garment, as does heat from a dryer, as well as direct UV light from the sun. You should have rec'd the special washing instructions for your medical garment. Usually I recommend that my patients wash on a gentle cycle using a simple detergent (woolite, or Ivory) without any bleach, double rinse it, and and then air dry it out of direct sunlight. Stain release agents may be used depending on the fabric if you have any blood or bodily fluid stains on it.
Regarding your other garments that you want to use instead, just bring them into your doctor's office and they can have a look at them and decide if they are appropriate in your case.