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.