That's not how it works. It's more like there's this shared network of participants that has strong incentives to consistently agree that you own a numeric measurement of "value" and that you have the agreed-upon ability to elect to transfer portions of this value to other participants in the network (because you _know_ a specific private key and can irrefutably demonstrate that you know it even without disclosing it).
Most of my finances are just strings of 1s and 0s. My company pays me. They don't send pieces of paper to my bank. It happens via 1s and 0s. From there, 80-90% of my expenses get paid automatically (rent, most taxes, electricity, internet, gas, water, trash collection) all via 1s and 0s. And cash is slowly disappearing into 1s and 0s as well via phone payment systems.
You don't, though. That's more like IP law, where people really do claim rights over the usage of strings. Crypto is more like allocating a string on some servers with a decentralized process
I don't think cryptocoins are about "claiming" ones and zeroes, but instead having the ability to "use" those ones and zeroes, unlike anybody else.
Although ultimately the effect is similar to "claiming" them by virtue of being the only one (as long as you don't share the private key of course) that can operate on those one and zeroes (i.e transfering them to some other address).
Unlike, say, copyright, where even if you claim ownership, someone else can also operate on it (e.g. piracy), illegally of course, but still can.
