No, they very explicitly don’t. They claim a licence in perpetuity to nearly all the same rights as the data owner, but the data subject is still the owner.
Also, that licence may not be so robust. A judge should see that the website has no obligation to continue hosting the website, and they offer nothing in return for the data, so the perpetual licence is not a reasonable term in the contract and should be struck down to something the data subject can rescind. In some respects we do have this with “the right to be forgotten” and to have businesses delete your data, however the enforcement of this is sorely lacking.
Laws change over time, though. Everyone is the victim of this practice, so eventually the law should catch up.
What do you think owning is? If I pull your strings, you’re my puppet. Ownership is control.
Ironically your metaphor works really well as an example of how that’s not true - just because you’re allowed to pull the strings, doesn’t mean you’re the owner of the puppet
