You still own the phone, you just have a loan basically for the amount of the cost of the phone that you are paying off monthly (or can pay off fully at any time).
The trickier part is that carriers have started offering deals where they will subsidize part of the cost by giving you a monthly credit over the term of the loan, so to get all of that subsidized part you would want to stay with the carrier for that whole time period--basically just a different way of doing contracts in a sense with a subsidized phone, just as carriers have been doing all along before.
One of the things carriers that are now doing more than ever, Are trade in incentives. When I traded in my iPhone 6, they gave me nearly $500 towards my bill credit. Even though it wasn't worth that, it was still a nice deduction off my monthly bill. So I guess there is a trade off when carriers want the consumer to upgrade. It almost resembles how car dealerships operate in some ways. They want to sell new products and have the consumer Upgrade.