Yes you can be addicted to popsicles just like you can be addicted to any food, drink or just about anything else.
I would say that I'm addicted to popsicles because I have to have one everyday at least in the summer but sometimes when it's really hot out I'll eat around 12 of the popsicles everyday.
However we also need to eat other things because popsicles alone cannot give you nutrients, vitamins and minerals or fill up your stomach.
You cannot live off popsicles alone.
While popsicles are a great treat and can keep you hydrated the popsicles are not food but instead a liquid so all you would be doing is hydrating yourself and getting liquids in your body.
You also need actual food to survive such as vegetables, fruits and meats.
So you would not live long if you just ate popsicles.
You could only live on strictly popsicles for around 2 to 4 weeks at most because you would still starve to death.
So if all you're doing is eat popsicles you will still be hungry and need to eat other foods because popsicles are just a liquid that is frozen to make the popsicles and not an actual food that provides any nutrients, vitamins, minerals etc that our bodies need to thrive.