Your post is very dangerous from a security perspective and may misinform many people who are still learning about digital identification.
> the role of proof of work is to establish a somewhat strange kind of identity among anonymous participants.
The role of proof of work is to make the creation of information one-way based upon a pre-specified amount of work. Therefore, in a properly constructed system, a participant cannot steal the results created by another party.
Identification is generally accepted in the security community to come from something you are (biometrics), something you have (bank card), and something you know (pin for your bank card, password, etc). For safety, two separate classes are normally and simultaneously required for identification, hence the term two-factor id.
An identity is something entirely different from work. Proof of work, since it is one-way, proves that work was performed. This has nothing to do with identity, which cannot be proved by work.
Something You Can Do is not a new category of identification but a statement about processing power. This can be seen by simply sharing the private key of a wallet. In two factor identification, you could make a copy of someone's eye or face to try to get into a system, but two factor identification would stop you.
> And this is what proof of work does, it limits your ability to invent identities and cast as many votes as you would like by making casting a vote really hard respectively expensive.
Proof of work doesn't have anything to do with the ability to invent identities. It ties the result of processing power to keys. Digital keys are not identities any more than physical keys are houses. To construct a house, you need to perform the work. To get into a house, you need a key. To take possession of a key you need to identify yourself.
> a 51 % attack is just someone using a lot of money to buy more than half of all the available identities in the Bitcoin system granting him the majority of the available votes.
A 51% attack is when someone who may or may not use a lot of money forces an alternate version of a blockchain data structure by having 51% or more of the processing power. This has nothing to do with purchasing identities. You can test this for yourself. If you create two bitcoin wallets, you clearly didn't create two digital identities. You are the identity in possession of two keys for two separate wallets.
Besides, a 51% attack isn't 51% of the nodes but 51% of the processing power, which could be only 1 powerful node. Those are entirely different.