In very much the same train of thought as in the necessity of intermediary banks, the property is yours the moment you pay for it. However, since the transaction didn't occur in person, it may very well be the property of the seller with the obligation to deliver the goods.
My wording may be a bit off, I'm no expert. But this is my interpretation of the matter.
I personally have a strict sense of "ownership". The easy proxy for it is: "Is there open firmware developed in FLOSS that I can replace the existing with?"
Obviously, that's not a 100%. Perhaps something like this is better:
"The thing is owned by me, when I pay for it, and it doesnt have irrevocable tie-backs to the company." Now, my strategy here isn't even foolproof, with auutomatic updates that can reverse features. Ideally, even automatic updates are a trap UNLESS you have a way to roll them back.
That's why I highly prefer hardware with FLOSS options. That's really the only way to guarantee my "ownership".
On a general level, depending on the incoterms. Specifically, it depends on whatever contract and term and conditions you agreed to as a customer. And whether or not those hold up to legal scrutiny.
Ownership is surprisingly well defined legally. And the general principle is actually quite easy to understand. The specifics depend. Hint: The police is only indirectly involved, if at all.
Property rights are not a binary, all or nothing thing, so the answer is complicated.
When do you have the right to use something? When do you have the exclusive right to use it? When do you have the right to eject others? When do you have the right to destroy it? Copy it? Sell it?
Some of these rights you may never have. Some you have the moment you pay, others are practically impossible to enforce until you take delivery.
My wording may be a bit off, I'm no expert. But this is my interpretation of the matter.