Definitely not. Drug addictions create a biochemical change; oxygen does not cause a biochemical change, so it cannot be addictive.
If you use home oxygen, you suffer from a physical barrier in your lungs. This barrier reduces the flow of oxygen gas from the lungs into your bloodstream. Home oxygen therapy simply increases the amount of oxygen in each breath compared to other gases.
Oxygen therapy patients typically need increasing amounts of oxygen over time, because their lung disease worsens. If you need additional oxygen, it is because of the progression of your condition, not because you are addicted to oxygen.