How do you get to own your fashion picks?

You own the items you buy once. However, in every challenge except for the daily challenge there is a point/value meter you have to fill up in order to submit a look. When you've bought an item once it loses it's value/points, but you can still use it in every look. I use my owned items to complete looks and make them look good and buy new items or use rewarded items to fill up the required point/value amount for each look. If you want to use an owned item and get the points/value for that item, you can "rebuy" the item and pay for it again to get the points for that specific look. I only do this if i'm low on money and the cheapest option is to rebuy my owned item instead of buying a new one. Otherwise, i don't recommend doing this. Hope this helps and explains things a little for you. I struggles with the economy of the game in the beginning as well and they're not very good at explaining it to you either, but with some time you'll learn and things will get easier.